Research Article Time Reversal of Volterra Processes...

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2013 Article ID 790709 13 pageshttpdxdoiorg1011552013790709

Research ArticleTime Reversal of Volterra Processes Driven StochasticDifferential Equations

L Decreusefond

Institut Telecom-Telecom ParisTech-CNRS LTCI 75013 Paris France

Correspondence should be addressed to L Decreusefond laurentdecreusefondtelecom-paristechfr

Received 18 June 2012 Accepted 27 December 2012

Academic Editor Ciprian A Tudor

Copyright copy 2013 L Decreusefond This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider stochastic differential equations driven by some Volterra processes Under time reversal these equations aretransformed into past-dependent stochastic differential equations driven by a standard Brownian motion We are then in positionto derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning

1 Introduction

Fractional Brownian motion (fBm for short) of Hurst index119867 isin [0 1] is the Gaussian process which admits the followingrepresentation for any 119905 ge 0

119861119867(119905) = int

119905

0

119870119867(119905 119904) d119861 (119904) (1)

where 119861 is a one-dimensional Brownian motion and 119870119867is a

triangular kernel that is119870119867(119905 119904) = 0 for 119904 gt 119905 the definition

of which is given in (46) Fractional Brownian motion isprobably the first process which is not a semimartingale andfor which it is still interesting to develop a stochastic calculusThat means we want to define a stochastic integral and solvestochastic differential equations driven by such a processFrom the very beginning of this program two approaches doexist One approach is based on the Holder continuity or thefinite 119901 variation of the fBm sample paths The other way toproceed relies on the gaussianity of fBmThe former ismainlydeterministic and was initiated by Zahle [1] Feyel and de laPradelle [2] and Russo and Vallois [3 4] Then came thenotion of rough paths was introduced by Lyons [5] whoseapplication to fBm relies on the work of Coutin and Qian [6]These works have been extended in the subsequent works [7ndash17] A new way of thinking came with the independent butrelated works of Feyel de la Pradelle [18] and Gubinelli [19]The integral with respect to fBm was shown to exist as theunique process satisfying some characterization (analytic in

the case of [18] algebraic in [19]) As a byproduct this showedthat almost all the existing integrals throughout the literaturewere all the same as they all satisfy these two conditionsBehind each approach but the last too is a constructionof an integral defined for a regularization of fBm thenthe whole work is to show that under some convenienthypothesis the approximate integrals converge to a quantitywhich is called the stochastic integral with respect to fBmThe main tool to prove the convergence is either integrationby parts in the sense of fractional deterministic calculus orenrichment of the fBm by some iterated integrals proved toexist independently or by analytic continuation [20 21]

In the probabilistic approach [22ndash30] the idea is also todefine an approximate integral and then prove its conver-gence It turns out that the key tool is here the integration byparts in the sense of Malliavin calculus

In dimension greater than one with the deterministicapproach one knows how to define the stochastic integral andprove existence and uniqueness of fBm-driven SDEs for fBmwith Hurst index greater than 14 Within the probabilisticframework one knows how to define a stochastic integral forany value of119867but one cannot prove existence anduniquenessof SDEs whatever the value of119867 The primary motivation ofthis work is to circumvent this problem

In [26 27] we defined stochastic integrals with respectto fBm as a ldquodamped-Stratonovitchrdquo integral with respectto the underlying standard Brownian motion This integralis defined as the limit of Riemann-Stratonovitch sums

2 International Journal of Stochastic Analysis

the convergence of which is proved after an integration byparts in the sense of Malliavin calculus Unfortunately thismanipulation generates nonadaptiveness formally the resultcan be expressed as

int

119905

0

119906 (119904) ∘ d119861119867 (119904) = 120575 (Klowast

119905119906) + trace (Klowast

119905nabla119906) (2)

whereK is defined by

K119891 (119905) =119889

119889119905int

119905

0

119870119867 (119905 119904) 119891 (119904) d119904 (3)

and Klowast

119905is the adjoint of K in L2

([0 119905]R) In particularthere exists 119896 such that

Klowast

119905119891 (119904) = int

119905

119904

119896 (119905 119906) 119891 (119906) d119906 (4)

for any 119891 isin L2([0 119905]R) so that even if 119906 is adapted (with

respect to the Brownian filtration) the process (119904 997891rarrKlowast

119905119906(119904))

is anticipative However the stochastic integral process (119905 997891rarrint119905

0119906(119904) ∘ d119861119867(119904)) remains adapted hence the anticipative

aspect is in some sense artificialThemotivation of this workis to show that up to time reversal we can work with adaptedprocess and Ito integralsThe time-reversal properties of fBmwere already studied in [31] in a different context It wasshown there that the time reversal of the solution of an fBm-driven SDE of the form

119889119884 (119905) = 119906 (119884 (119905)) d119905 + d119861119867 (119905) (5)

is still a process of the same form With a slight adaptationof our method to fBm-driven SDEs with drift one shouldrecover the main theorem of [31]

In what follows there is no restriction on the dimensionbut we need to assume that any component of 119861119867 is an fBm ofHurst index greater than 12 Consider that we want to solvethe following equation

119883119905= 119909 + int

119905

0

120590 (119883119904) ∘ d119861119867 (119904) 0 le 119905 le 119879 (6)

where 120590 is a deterministic function whose properties will befixed below It turns out that it is essential to investigate themore general equations

119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119861119867 (119904) 0 le 119903 le 119905 le 119879 (A1015840

)

The strategy is then as follows We will first consider thereciprocal problem

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119861119867 (119904) 0 le 119903 le 119905 le 119879 (B1015840)

The first critical point is that when we consider 119885119903119905=

119884119905minus119903119905

119903 isin [0 119905] this process solves an adapted past-dependent and stochastic differential equation with respectto a standard Brownian motion Moreover because 119870

119867

is lower-triangular and sufficiently regular the trace term

vanishes in the equation defining 119885 We have then reducedthe problem to an SDE with coefficients dependent on thepast a problem which can be handled by the usual contrac-tion methods We do not claim that the results presentedare new (for instance see the brilliant monograph [32] fordetailed results obtained via rough paths theory) but it seemsinteresting to have purely probabilistic methods which showthat fBm driven SDEs do have strong solutions which arehomeomorphisms Moreover the approach given here showsthe irreducible difference between the case119867 lt 12 and119867 gt

12 The trace term only vanishes in the latter situation sothat such an SDE is merely a usual SDE with past-dependentcoefficients This representation may be fruitful for instanceto analyze the support and prove the absolute continuity ofsolutions of (6)

This paper is organized as follows After some preliminar-ies on fractional Sobolev spaces often called Besov-Liouvillespace we address in Section 3 the problem of Malliavincalculus and time reversal This part is interesting in its ownsince stochastic calculus of variations is a framework oblivi-ous to time Constructing such a notion of time is achievedusing the notion of resolution of the identity as introduced in[33 34] We then introduce the second key ingredient whichis the notion of strict causality or quasinilpotence see [35]for a related application In Section 4 we show that solving(B1015840) reduces to solve a past-dependent stochastic differentialequation with respect to a standard Brownian motion see(C) below Then we prove existence uniqueness and someproperties of this equation Technical lemmas are postponedto Section 5

2 Besov-Liouville Spaces

Let 119879 gt 0 be fix real number For a measurable function 119891 [0 119879] rarr R119899 we define 120591

119879119891 by

120591119879119891 (119904) = 119891 (119879 minus 119904) for any 119904 isin [0 119879] (7)

For 119905 isin [0 119879] 119890119905119891 will represent the restriction of 119891 to [0 119905]

that is 119890119905119891 = 1198911

[0119905] For any linear map 119860 we denote by 119860lowast

119879

its adjoint in L2([0 119879]R119899) For 120578 isin (0 1] the space of 120578-

Holder continuous functions on [0 119879] is equipped with thenorm

10038171003817100381710038171198911003817100381710038171003817Hol(120578) = sup

0lt119904lt119905lt119879

1003816100381610038161003816119891 (119905) minus 119891 (119904)1003816100381610038161003816

|119905 minus 119904|120578

+10038171003817100381710038171198911003817100381710038171003817infin (8)

Its topological dual is denoted by Hol(120578)lowast For 119891 isin

L1([0 119879] R119899 d119905) (denoted by L1 for short) the left and

right fractional integrals of 119891 are defined by

(119868120574

0+119891) (119909) =

1

Γ (120574)int

119909

0

119891 (119905) (119909 minus 119905)120574minus1d119905 119909 ge 0

(119868120574

119879minus119891) (119909) =

1

Γ (120574)int

119879

119909

119891 (119905) (119905 minus 119909)120574minus1d119905 119909 le 119879

(9)

International Journal of Stochastic Analysis 3

where 120574 gt 0 and 11986800+ = 119868

0

119879minus = Id For any 120574 ge 0 119901 119902 ge 1 any

119891 isinL119901 and 119892 isinL119902 where 119901minus1 + 119902minus1 le 120574 we have

int

119879

0

119891 (119904) (119868120574

0+119892) (119904) d119904 = int

119879

0

(119868120574

119879minus119891) (119904) 119892 (119904) d119904 (10)

The Besov-Liouville space 1198681205740+(L119901

) = I+

120574119901is usually

equipped with the norm10038171003817100381710038171003817119868120574

0+11989110038171003817100381710038171003817I+120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(11)

Analogously the Besov-Liouville space 119868120574119879minus(L119901

) = Iminus

120574119901

is usually equipped with the norm10038171003817100381710038171003817119868minus120574

119879minus11989110038171003817100381710038171003817Iminus120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(12)

We then have the following continuity results (see [2 36])

Proposition 1 Consider the following(i) If 0 lt 120574 lt 1 1 lt 119901 lt 1120574 then 119868120574

0+is a bounded

operator fromL119901 intoL119902 with 119902 = 119901(1 minus 120574119901)minus1(ii) For any 0 lt 120574 lt 1 and any 119901 ge 1 I+

120574119901is continuously

embedded in119867119900119897(120574 minus 1119901) provided that 120574 minus 1119901 gt 0(iii) For any 0 lt 120574 lt 120573 lt 1 Hol (120573) is compactly

embedded inI120574infin

(iv) For 120574119901 lt 1 the spaces I+

120574119901and Iminus

120574119901are canonically

isomorphic We will thus use the notation I120574119901

todenote any of these spaces

3 Malliavin Calculus and Time Reversal

Our reference probability space is Ω = C0([0 119879]R119899) the

space of R119899-valued continuous functions null at time 0 TheCameron-Martin space is denoted by H and is defined asH = 119868

1

0+(L

2([0 119879])) In what follows the space L2

([0 119879])

is identified with its topological dual We denote by 120581

the canonical embedding from H into Ω The probabilitymeasure P on Ω is such that the canonical map 119882

120596 997891rarr (120596(119905) 119905 isin [0 119879]) defines a standard 119899-dimensionalBrownian motion A mapping 120601 fromΩ into some separableHilbert space H is called cylindrical if it is of the form120601(119908) = sum

119889

119894=1119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩)119909

119894 where for each 119894119891

119894isin

Cinfin

0(R119899R) and (119907

119894119895 119895 = 1 119899) is a sequence of Ωlowast For

such a function we define nablaW120601 as

nablaW120601 (119908) = sum

119894119895=1

120597119895119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩) 119907

119894119895otimes 119909

119894

(13)

where 119907 is the image of 119907 isin Ωlowast by the map (11986810+ ∘ 120581)

lowast Fromthe quasi-invariance of the Wiener measure [37] it followsthat nablaW is a closable operator on 119871119901(ΩH) 119901 ge 1 and wewill denote its closure with the same notation The powers ofnablaW are defined by iterating this procedure For 119901 gt 1 119896 isin N

we denote byD119901119896(H) the completion ofH-valued cylindrical

functions under the following norm

10038171003817100381710038171206011003817100381710038171003817119901119896

=

119896

sum

119894=0

10038171003817100381710038171003817(nabla

W)11989412060110038171003817100381710038171003817119871119901(ΩHotimesL119901([01])otimes119894)

(14)

We denote by L1199011

the space D1199011(L119901

([0 119879] R119899)) Thedivergence denoted as 120575W is the adjoint of nablaW 119907 belongsto Dom

119901120575W whenever for any cylindrical 120601

100381610038161003816100381610038161003816100381610038161003816

E [int119879

0

119907119904nablaW119904120601d119904]

100381610038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171206011003817100381710038171003817119871119901

(15)

and for such a process 119907

E [int119879

0

119907119904nablaW119904120601d119904] = E [120601 120575W119907] (16)

We introduced the temporary notation 119882 for standardBrownian motion to clarify the forthcoming distinctionbetween a standard Brownian motion and its time reversalActually the time reversal of a standard Brownian is also astandard Brownian motion and thus both of them ldquoliverdquo inthe sameWiener space We now precise how their respectiveMalliavin gradient and divergence are linked Consider 119861 =(119861(119905) 119905 isin [0 119879]) an 119899-dimensional standard Brownianmotion and 119879 = (119861(119879)minus119861(119879minus119905) 119905 isin [0 119879]) its time reversalConsider the following map

Θ119879 Ω 997888rarr Ω

120596 997891997888rarr = 120596 (119879) minus 120591119879120596

(17)

and the commutative diagram

ℒ2

ℒ2

120591119879

Ω sup H H sub ΩΘ119879

1198681

0+ 119868

1

0+

(18)

Note that Θminus1

119879= Θ

119879since 120596(0) = 0 For a function 119891 isin

Cinfin

119887(R119899119896) we define the following

nabla119903119891 (120596 (119905

1) 120596 (119905

119896))

=

119896

sum

119895=1

120597119895119891 (120596 (119905

1) 120596 (119905

119896)) 1

[0119905119895] (119903)

nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[0119905119895] (119903)

(19)

The operator nabla = nabla119861 (resp nabla = nabla

) is the Malliavingradient associated with a standard Brownian motion (respits time reversal) Since

119891 ( (1199051) (119905

119896))

= 119891 (120596 (119879) minus 120596 (119879 minus 1199051) 120596 (119879) minus 120596 (119879 minus 119905119896))

(20)


we can consider 119891((1199051) (119905

119896)) as a cylindrical function

with respect to the standard Brownian motion As such itsgradient is given by

nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[119879minus119905119895119879] (119903)

(21)

We thus have for any cylindrical function 119865

nabla119865 ∘ Θ119879 (120596) = 120591119879nabla119865 () (22)

Since Θlowast

119879P = P and 120591

119879is continuous from L119901 into itself

for any 119901 it is then easily shown that the spacesD119901119896

and D119901119896

(with obvious notations) coincide for any 119901 119896 and that (22)holds for any element of one of these spaces Hence we haveproved the following theorem

Theorem 2 For any 119901 ge 1 and any integer 119896 the spaces D119901119896

and D119901119896

coincide For any 119865 isin D119901119896

for some 119901 119896

nabla (119865 ∘ Θ119879) = 120591

119879nabla (119865 ∘ Θ

119879) P as (23)

By duality an analog result follows for divergences

Theorem 3 A process 119906 belongs to the domain of 120575 if andonly if 120591

119879119906 belongs to the domain of 120575 and then the following

equality holds

120575 (119906 ()) () = 120575 (120591119879119906 ()) (120596) = 120575 (120591119879119906 ∘ Θ119879) (120596) (24)

Proof For119906 isinL2 for cylindrical119865 we have on the one hand

E [119865 () 120575119906 ()] = E [(nabla119865 () 119906)L2] (25)

and on the other hand

E [(nabla119865 () 119906)L2] = E [(120591

119879nabla119865 ∘ Θ

119879 (120596) 119906)L2]

= E [(nabla119865 ∘ Θ119879 (120596) 120591119879119906)L2

]

= E [119865 ∘ Θ119879 (120596) 120575 (120591119879119906) (120596)]

= E [119865 () 120575 (120591119879119906) (120596)]

(26)

Since this is valid for any cylindrical 119865 (24) holds for 119906 isinL2Now for 119906 in the domain of divergence (see [37 38])

120575119906 = sum

119894

((119906 ℎ119894)L2120575ℎ

119894minus (nabla119906 ℎ

119894otimes ℎ

119894)L2otimesL2

) (27)

where (ℎ119894 119894 isin N) is an orthonormal basis ofL2

([0 119879] R119899)Thus we have

120575 (119906 ()) () = sum

119894

((119906 () ℎ119894)L2120575ℎ119894 ()

minus(nabla119906 () ℎ119894 otimes ℎ119894)L2otimesL2)

= sum

119894

((119906 () ℎ119894)L2120575 (120591

119879ℎ119894) (120596)

minus(nabla119906 () 120591119879ℎ119894 otimes ℎ119894)L2otimesL2)

= sum

119894

((120591119879119906 () 120591119879ℎ119894)L2

120575 (120591119879ℎ119894) (120596)

minus(nabla120591119879119906 () 120591119879ℎ119894 otimes 120591119879ℎ119894)L2otimesL2

)

(28)

where we have taken into account that 120591119879is in an involution

Since (ℎ119894 119894 isin N) is an orthonormal basis of L2

([0 119879] R119899)identity (24) is satisfied for any 119906 in the domain of 120575

31 Causality and Quasinilpotence In anticipative calculusthe notion of trace of an operator plays a crucial roleWe referto [39] for more details on trace

Definition 4 Let 119881 be a bounded map from L2([0 119879] R119899)

into itself The map 119881 is said to be trace class whenever forone CONB (ℎ

119899 119899 ge 1) ofL2

([0 119879] R119899)

sum

119899ge1

1003816100381610038161003816(119881ℎ119899 ℎ119899)L21003816100381610038161003816 is finite (29)

Then the trace of 119881 is defined by

trace (119881) = sum119899ge1

(119881ℎ119899 ℎ

119899)L2 (30)

It is easily shown that the notion of trace does not dependon the choice of the CONB

Definition 5 A family 119864 of projections (119864120582 120582 isin [0 1])

in L2([0 119879] R119899) is called a resolution of the identity if it

satisfies the conditions

(1) 1198640= 0 and 119864

1= Id

(2) 119864120582119864120583= 119864

120582and120583

(3) lim120583darr120582119864120583= 119864

120582for any 120582 isin [0 1) and lim

120583uarr1119864120583= Id

For instance the family 119864 = (119890120582119879 120582 isin [0 1]) is a

resolution of the identity inL2([0 119879] R119899)

Definition 6 A partition 120587 of [0 119879] is a sequence 0 = 1199050lt

1199051lt sdot sdot sdot lt 119905

119899= 119879 Its mesh is denoted by |120587| and defined by

|120587| = sup119894|119905119894+1minus 119905

119894|

The causality plays a crucial role in what followsThe nextdefinition is just the formalization in terms of operator of theintuitive notion of causality


Definition 7 A continuous map 119881 from L2([0 119879] R119899) into

itself is said to be 119864 causal if and only if the followingcondition holds

119864120582119881119864

120582= 119864

120582119881 for any 120582 isin [0 1] (31)

For instance an operator 119881 in integral form 119881119891(119905) =

int119879

0119881(119905 119904)119891(119904)d119904 is causal if and only if 119881(119905 119904) = 0 for 119904 ge 119905

that is computing119881119891(119905) needs only the knowledge of 119891 up totime 119905 and not after Unfortunately this notion of causality isinsufficient for our purpose and we are led to introduce thenotion of strict causality as in [40]

Definition 8 Let 119881 be a causal operator It is a strictly causaloperator whenever for any 120576 gt 0 there exists a partition 120587 of[0 119879] such that for any 1205871015840 = 0 = 119905

0lt 119905


119899= 119879 sub 120587

10038171003817100381710038171003817(119864

119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)10038171003817100381710038171003817L2

lt 120576 for 119894 = 0 119899 minus 1(32)

Note carefully that the identity map is causal but notstrictly causal Indeed if 119881 = Id for any 119904 lt 119905

1003817100381710038171003817(119864119905 minus 119864119904)119881(119864119905 minus 119864119904)1003817100381710038171003817L2

=1003817100381710038171003817119864119905 minus 119864119904

1003817100381710038171003817L2= 1 (33)

since 119864119905minus 119864

119904is a projection However if 119881 is hyper-contract-

ive we have the following result

Lemma 9 Assume the resolution of the identity to be either119864 = (119890

120582119879 120582 isin [0 1]) or 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) If 119881 is

an 119864 causal map continuous fromL2 intoL119901 for some 119901 gt 2then 119881 is strictly 119864 causal

Proof Let 120587 be any partition of [0 119879] Assume that 119864 =

(119890120582119879 120582 isin [0 1]) and the very same proof works for the other

mentioned resolution of the identity According to Holderformula we have for any 0 le 119904 lt 119905 le 119879

1003817100381710038171003817(119864119905 minus 119864119904) 119881 (119864119905 minus 119864119904) 1198911003817100381710038171003817L2

= int

119905

119904

1003816100381610038161003816119881(1198911(119904119905])(119906)1003816100381610038161003816

2d119906

le (119905 minus 119904)1minus21199011003817100381710038171003817119881(1198911(119904119905])

1003817100381710038171003817L1199012

le 119888 (119905 minus 119904)1minus21199011003817100381710038171003817119891

1003817100381710038171003817L2

(34)

Then for any 120576 gt 0 there exists 120578 gt 0 such that |120587| lt 120578

implies (119864119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)119891L2 le 120576 for any 0 = 1199050 lt1199051lt sdot sdot sdot lt 119905

119899= 119879 sub 120587 and any 119894 = 0 119899 minus 1

The importance of strict causality lies in the next theoremwe borrow from [40]

Theorem 10 The set of strictly causal operators coincides withthe set of quasinilpotent operators that is trace-class operatorssuch that trace(119881119899) = 0 for any integer 119899 ge 1

Moreover we have the following stability theorem

Theorem 11 The set of strictly causal operators is a two-sidedideal in the set of causal operators

Definition 12 Let 119864 be a resolution of the identity inL2([0 119879]R119899) Consider the filtrationF119864 defined as

F119864

119905= 120590 120575

W(119864

120582ℎ) 120582 le 119905 ℎ isinL

2 (35)

An L2-valued random variable 119906 is said to be F119864 adaptedif for any ℎ isin L2 the real valued process ⟨119864

120582119906 ℎ⟩ is F119864-

adaptedWedenote byD119864

119901119896(H) the set ofF119864 adapted random

variables belonging to D119901119896(H)

If 119864 = (119890120582119879 120582 isin [0 1]) the notion of F119864 adapted proc-

esses coincides with the usual one for the Brownian filtrationand it is well known that a process 119906 is adapted if and only ifnablaW119903119906(119904) = 0 for 119903 gt 119904 This result can be generalized to any

resolution of the identity

Theorem 13 (Proposition 31 of [33]) Let 119906 belongs to L1199011

Then 119906 isF119864 adapted if and only if nablaW119906 is 119864 causal

We then have the following key theorem

Theorem 14 Assume the resolution of the identity to be 119864 =(119890120582119879 120582 isin [0 1]) either 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) and that

119881 is an 119864-strictly causal continuous operator fromL2 intoL119901

for some 119901 gt 2 Let 119906 be an element of D119864

21(L2

) Then 119881nablaW119906

is of trace class and we have trace(119881nablaW119906) = 0

Proof Since 119906 is adapted nablaW119906 is 119864-causal According to

Theorem 11 119881nablaW119906 is strictly causal and the result follows by

Theorem 10

In what follows 1198640 is the resolution of the identity inthe Hilbert space L2 defined by 119890

120582 119879119891 = 1198911

[0120582119879]and 0 is

the resolution of the identity defined by 119890120582119879119891 = 1198911

[(1minus120582)119879119879]

The filtrations F1198640

and F0

are defined accordingly Nextlemma is immediate when 119881 is given in the form of 119881119891(119905) =int119905

0119881(119905 119904)119891(119904)d119904 Unfortunately such a representation as an

integral operator is not always available We give here analgebraic proof to emphasize the importance of causality

Lemma 15 Let 119881 be a map from L2([0 119879] R119899) into itself

such that 119881 is 1198640-causal Let 119881lowast be the adjoint of 119881 inL2([0 119879]R119899) Then the map 120591

119879119881lowast

119879120591119879is 0-causal

Proof This is a purely algebraic lemma once we have noticedthat

120591119879119890119903= (Id minus 119890

119879minus119903) 120591

119879for any 0 le 119903 le 119879 (36)

For it suffices to write

120591119879119890119903119891 (119904) = 119891 (119879 minus 119904) 1[0119903] (119879 minus 119904)

= 119891 (119879 minus 119904) 1[119879minus119903119879] (119904)

= (Id minus 119890119879minus119903) 120591

119879119891 (119904) for any 0 le 119904 le 119879

(37)


We have to show that

119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890

119903and 120591lowast

119879= 120591

119879 Now (37) yields

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)

Use (37) again to obtain

119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)

since 119881 is 119864-causal

32 Stratonovitch Integrals In what follows 120578 belongs to(0 1] and 119881 is a linear operator For any 119901 ge 2 we set thefollowing

Hypothesis 1 (119901 120578) The linear map 119881 is continuous fromL119901

([0 119879]R119899) into the Banach space Hol(120578)

Definition 16 Assume that Hypothesis 1 (119901 120578) holds TheVolterra process associated to 119881 denoted by119882119881 is definedby

119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)

For any subdivision 120587 of [0 119879] that is 120587 = 0 = 1199050lt


119899= 119879 of mesh |120587| we consider the Stratonovitch

sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)

Definition 17 We say that 119906 is 119881-Stratonovitch integrable on[0 119905]whenever the family 119877120587(119905 119906) defined in (43) convergesin probability as |120587| goes to 0 In this case the limit will bedenoted by int119905

0119906(119904) ∘ d119882119881

(119904)

Example 18 Thefirst example is the so-called Levy fractionalBrownian motion of Hurst index119867 gt 12 defined as

1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)

This amounts to say that 119881 = 119868119867minus12

119879minus

ThusHypothesis 1 (119901119867 minus 12 minus 1119901) holds provided that119901(119867 minus 12) gt 1

Example 19 The other classical example is the fractionalBrownian motion with stationary increments of Hurst index119867 gt 12 which can be written as

int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1

2minus 119867119867 minus

1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)

The Gauss hypergeometric function 119865(120572 120573 120574 119911) (see [41]) isthe analytic continuation on C times C times C minus1 minus2 times 119911 isin

C Arg|1 minus 119911| lt 120587 of the power series

+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)

Γ (119886)= 119886 (119886 + 1) sdot sdot sdot (119886 + 119896 minus 1)

(47)

We know from [36] that 119870119867

is an isomorphism from L119901

([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)

Consider thatK119867= 119868

minus1

0+ ∘ 119870119867 Then it is clear that

int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)

hence we are in the framework of Definition 17 provided thatwe take 119881 = (K

119867)lowast

119879 Hypothesis 1 (119901119867 minus 12 minus 1119901) is

satisfied provided that 119901(119867 minus 12) gt 1

The next theorem then follows from [26]

Theorem 20 Assume that Hypothesis 1 (119901 120578) holds Assumethat 119906 belongs toL

1199011Then 119906 is119881-Stratonovitch integrable and

there exists a process which we denote by 119863W119906 such that 119863W

119906

belongs to 119871119901(P otimes 119889119904) and

int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)

The so-called ldquotrace-termrdquo satisfies the following estimate

E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)

for some universal constant 119888 Moreover for any 119903 le 119879 119890119903119906 is

119881-Stratonovitch integrable and

int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)


and we have the maximal inequality

E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)

where 119888 does not depend on 119906

The main result of this Section is the following theoremwhich states that the time reversal of a Stratonovitch integralis an adapted integral with respect to the time-reversedBrownian motion Due to its length its proof is postponedto Section 51

Theorem 21 Assume that Hypothesis 1 (119901 120578) holds Let 119906belong to L

1199011and let

119879= 120591

119905119881120591

119879 Assume furthermore that

119881 is 0-causal and that = 119906 ∘ Θminus1

119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT

(s) 0 le r le t le T(54)

where the last integral is an Ito integral with respect to the timereversed Brownianmotion 119879(119904) = 119861(119879)minus119861(119879minus119904) = Θ

119879(119861)(119904)

Remark 22 Note that at a formal level we could have an easyproof of this theorem For instance consider the Levy fBmand a simple computation shows that

119879= 119868

119867minus12

0+

for any119879 Thus we are led to compute trace(119868119867minus12

0+

nabla119906) If we hadsufficient regularity we could write

trace (119868119867minus120+ nabla119906) = int

119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)

since nabla119904119906(119903) = 0 for 119904 gt 119903 for 119906 adapted Obviously there are

many flaws in these lines of proof The operator 119868119867minus120+

nabla119906 isnot regular enough for such an expression of the trace to betrue Even more there is absolutely no reason for

119879nabla119906 to be

a kernel operator so we cannot hope such a formula Theseare the reasons that we need to work with operators and notwith kernels

4 Volterra-Driven SDEs

LetG be the group of homeomorphisms ofR119899 equipped withthe distance We introduce a distance 119889 onG by

119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)

Then G is a complete topological group Consider theequations

119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)

As a solution of (A) is to be constructed by ldquoinvertingrdquoa solution of (B) we need to add to the definition of asolution of (A) or (B) the requirement of being a flowof homeomorphisms This is the meaning of the followingdefinition

Definition 23 By a solution of (A) we mean a measurablemap

Ω times [0 119879] times [0 119879] 997888rarr G

(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)

such that the following properties are satisfied

(1) For any 0 le 119903 le 119905 le 119879 for any 119909 isin R119899 119883119903119905(120596 119909) is

120590119882119881(119904) 119903 le 119904 le 119905-measurable

(2) For any 0 le 119903 le 119879 for any 119909 isin R119899 the processes(120596 119905) 997891rarr 119883

119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011

for some 119901 ge 2(3) For any 0 le 119903 le 119904 le 119905 for any 119909 isin R119899 the following

identity is satisfied

119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)

(4) Equation (A) is satisfied for any 0 le 119903 le 119905 le 119879 P-as

Definition 24 By a solution of (B) we mean a measurablemap


(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011

for some 119901 ge 2(3) Equation (B) is satisfied for any 0 le 119903 le 119905 le 119879 P-as(4) For any 0 le 119903 le 119904 le 119905 for any 119909 isin R119899 the following


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)

At last consider the equation for any 0 le 119903 le 119905 le 119879

119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)

where 119861 is a standard 119899-dimensional Brownian motion


Definition 25 By a solution of (C) we mean a measurablemap


(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable

(2) For any 0 le 119903 le 119905 le 119879 for any 119909 isin R119899 the processes(120596 119903) 997891rarr 119885

119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011

for some 119901 ge 2(3) Equation (C) is satisfied for any 0 le 119903 le 119905 le 119879P-as

Theorem 26 Assume that 119879is an 1198640 causal map continuous

from L119901 into I120572119901

for 120572 gt 0 and 119901 ge 4 such that 120572119901 gt 1Assume that 120590 is Lipschitz continuous and sublinear see (96)for the definition Then there exists a unique solution to (C)Let 119885 denote this solution For any (119903 1199031015840)

E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)

Since this proof needs several lemmas we defer it toSection 52

Theorem 27 Assume that 119879is an 1198640-causal map continuous

fromL119901 intoI120572119901

for 120572 gt 0 and 119901 ge 2 such that 120572119901 gt 1 Forfixed 119879 there exists a bijection between the space of solutionsof (B) on [0 119879] and the set of solutions of (C)

Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)

According to Theorem 21 119884 satisfies (B) if and only if 119885satisfies (C) The regularity properties are immediate sinceL119901 is stable by 120591

119879

The first part of the next result is then immediate

Corollary 28 Assume that 119879is an 1198640-causal map contin-

uous from L119901 into I120572119901

for 120572 gt 0 and 119901 ge 2 such that120572119901 gt 1 Then (B) has one and only one solution and for any0 le 119903 le 119904 le 119905 for any 119909 isin R119899 the following identity is satisfied

119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)

Proof According toTheorems 27 and 26 (B) has at most onesolution since (C) has a unique solution As to the existencepoints from (1) to (3) are immediately deduced from thecorresponding properties of 119885 and (66)

According to Theorem 26 (120596 119903) 997891rarr 119884119903119904(120596 119884

119904119905(120596 119909))

belongs to L1199011 hence we can apply the substitution formula

and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)

Then in view of (68) 119877 appears to be the unique solution (B)and thus 119877

119904119905(120596 119909) = 119884

119904119905(120596 119909) Point (4) is thus proved

Corollary 29 For 119909 fixed the random field (119884119903119905(119909) 0 le 119903 le

119905 le 119879) admits a continuous version Moreover

E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)

We still denote by 119884 this continuous version

Proof Without loss of generality assume that 119904 le 1199041015840 and

remark that 1198841199041199041015840(119909)

thus belongs to 120590119879119906 119906 ge 119904

E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119885119904minus119903119904 (119909) minus 119885119904minus1199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [1003816100381610038161003816119885119904minus1199031015840 119904 (119909) minus 119885119904minus1199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])

(71)

According toTheorem 37

E [1003816100381610038161003816119885119904minus119903119904(119909) minus 119885119904minus1199031015840 119904(119909)1003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)

In view of Theorem 21 the stochastic integral which appearsin (C) is also a Stratonovitch integral hence we can apply thesubstitution formula and say

119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)


Thus we can apply Theorem 37 and obtain that

E [1003816100381610038161003816119885119904minus1199031015840 119904(119909) minus 119885119904minus1199031015840 119904(1198841199041199041015840(119909))1003816100381610038161003816

119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)

The right hand side of this equation is in turn equal toE[|119885

01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get

E [1003816100381610038161003816119885119904minus1199031015840 119904 (119909) minus 119885119904minus1199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)

Combining (72) and (75) gives

E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)

hence the result comes

Thus we have the main result of this paper


fromL119901 intoI120572119901

for 120572 gt 0 and 119901 ge 4 such that 120572119901 gt 1 Then(A) has one and only one solution

Proof Under the hypothesis we know that (B) has a uniquesolution which satisfies (67) By definition a solution of (B)the process 119884minus1 (120596 119904) 997891rarr 119884

minus1

119904119905(120596 119909) belongs to L

1199011 hence

we can apply the substitution formula Following the lines ofproof of the previous theorem we see that119884minus1 is a solution of(A)

In the reverse direction two distinct solutions of (A)would give rise to two solutions of (B) by the same principlesSince this is definitely impossible in view of Corollary 28 (A)has at most one solution

5 Technical Proofs

51 Substitution Formula The proof of Theorem 21 relies onseveral lemmas including one known in anticipative calculusas the substitution formula compare [38]


1199011 If 119881nablaW

119906 is of trace class then

int

119879

0

119863W119906 (119904) ds = trace (VnablaWu) (77)

Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)

Proof For each 119896 let (120601119896119898 119898 = 1 2

119896) be the functions

120601119896119898

= 211989621

[(119898minus1)2minus1198961198982minus119896) Let 119875

119896be the projection onto the

span of the 120601119896119898

since nablaW119881119906 is of trace class we have (see

[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)

According to the proof of Theorem 20 the first part of thetheorem follows The second part is then a rewriting of (51)

For 119901 ge 1 let Γ119901be the set of random fields

119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)

equipped with the seminorms

119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)

for any compact119870 of R119898

Corollary 32 (substitution formula) Assume that Hypothesis1 (119901 120578) holds Let 119906(119909) 119909 isin R119898 belong to Γ

119901 Let 119865 be a

random variable such that ((120596 119904) 997891rarr 119906(120596 119904 119865)) belongs toL1199011

Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)

Proof Simple random fields of the form

119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L

1199011are dense in Γ

119901 In view of (53)

it is sufficient to prove the result for such random fields Bylinearity we can reduce the proof to randomfields of the form119867(119909)119906(120596 119904) Now for any partition 120587

120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)

According toTheorem 20 (83) is satisfied for simple randomfields

Definition 33 For any 0 le 119903 le 119905 le 119879 for 119906 in L1199011 we define

int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)

By the very definition of trace class operators the nextlemma is straightforward

Lemma 34 Let 119860 and 119861 be two continuous maps fromL2([0 119879] R119899) into itselfThen themap 120591

119879119860otimes119861 (resp119860120591

119879otimes119861)

is of trace class if and only if the map 119860 otimes 120591119879119861 (resp 119860 otimes 119861120591

119879)

is of trace class Moreover in such a situation

trace (120591119879119860 otimes 119861)

= trace (119860 otimes 120591119879119861) resp trace (119860120591

119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)

Thenext corollary follows by a classical density argument

Corollary 35 Let119906 isin L21

such thatnablaWotimes120591

119879119881119906 andnablaW

otimes119881120591119879119906

are of trace class Then 120591119879nablaWotimes119881119906 and nablaW

120591119879otimes119881119906 are of trace

class Moreover we have

trace (nablaWotimes 120591

119879119881119906) = trace (120591

119879nablaWotimes 119881119906)

trace (nablaWotimes (119881120591

119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)

Proof of Theorem 21 We first study the divergence term Inview of Theorem 3 we have

120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)

According to Lemma 15 (119879)lowast is

0causal and according to

Lemma 9 it is strictly 0causal Thus Theorem 14 implies

that nabla119881(119890119905minus 119890

119903) is of trace class and quasinilpotent Hence

Corollary 35 induces that

120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)

is trace class and quasinilpotent Now according toTheorem 2 we have

120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)

According toTheorem 20 we have proved (54)

52 The Forward Equation

Lemma 36 Assume that Hypothesis 1 (119901 120578) holds and that 120590is Lipschitz continuous Then for any 0 le 119886 le 119887 le 119879 the map

119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)

is Lipschitz continuous and Gateaux differentiable Its differen-tial is given by

119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)

Assume furthermore that 120590 is sublinear that is

|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)

Then for any 120595 isin 119862([0 119879]R119899) for any 119905 isin [0 119879]

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)

Proof Let 120595 and 120601 be two continuous functions since119862([0 119879]R119899) is continuously embedded inL119901 and

119879(120590 ∘120595minus

120590 ∘ 120601) belongs to Hol(120578) Moreover

sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)

since 120590 is Lipschitz continuousLet 120601 and 120595 be two continuous functions on [0 119879] Since

120590 is Lipschitz continuous we have

120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)


Moreover since 120590 is Lipschitz 1205901015840 is bounded and

int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)

This means that (119905 997891rarr int1

01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs

toL119901 Hence according to Hypothesis 1100381710038171003817100381710038171003817100381710038171003817

119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)

and 119879∘120590 is Gateaux differentiable and its differential is given

by (95)Since 120590∘120595 belongs to119862([0 119879]R119899) according to Hypoth-

esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)

The proof is thus complete

Following [43] we then have the following nontrivialresult

Theorem 37 Assume that Hypothesis 1 (119901 120578) holds and that120590 is Lipschitz continuous Then there exists one and only onemeasurable map fromΩtimes [0 119879] times [0 119879] intoG which satisfiesthe first two points of Definition (C) Moreover

E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)

and for any 119909 isin R119899 for any 0 le 119903 le 119905 le 119879 we have

E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)

Note even if 119909 and 1199091015840 are replaced by 120590119879(119906) 119905 le 119906

measurable random variables the last estimates still hold

Proof Existence uniqueness and homeomorphy of a solu-tion of (C) follow from [43] The regularity with respectto 119903 and 119909 is obtained as usual by BDG inequality andGronwall Lemma For 119909 or 1199091015840 random use the independenceof 120590119879(119906) 119905 le 119906 and 120590119879(119906) 119903 and 1199031015840 le 119906 le 119905

Theorem 38 Assume that Hypothesis 1 (119901 120578) holds and that120590 is Lipschitz continuous and sublinear Then for any 119909 isin R119899for any 0 le 119903 le 119904 le 119905 le 119879 (120596 119903) 997891rarr 119885

119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011

Proof According to ([44] Theorem 31) the differentiabilityof 120596 997891rarr 119885

119903119905(120596 119909) is ensured Furthermore

nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)

where 1205901015840 is the differential of 120590 For119872 gt 0 let

120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)

Since 119879is continuous fromL119901 into itself and 120590 is Lipschitz

according to BDG inequality for 119903 le 119906

E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)

Then Gronwall Lemma entails that

E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)

hence by Fatou lemma

E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)

The integrability of E[|nabla119906119885119903119905|119901] with respect to 119906 follows

Now since 0 le 119903 le 119904 le 119905 le 119879 119885119904119905(119909) is independent

of 119885119903119904(119909) thus the previous computations still hold and

(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011


According to [45] to prove that 119885minus1119903119905(119909) belongs to D

1199011

we need to prove that

(1) for every ℎ isin L2 there exists an absolutely continu-ous version of the process (119905 997891rarr 119885

minus1

119903119905(120596 + 119905ℎ 119909))

(2) there exists 119863119885minus1119903119905 an L2-valued random variable

such that for every ℎ isinL2

1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)

where the convergence holds in probability(3) 119863119885minus1

119903119905belongs toL2

(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup

119906le119907|120597119909119885119906119905(119909)|The same kind of computations as

above entails that (for the sake of brevity we do not detail thelocalisation procedure as it is similar to the previous one)

E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)

and (112) follows by Fatou and Gronwall lemmas Since119885119903119905(120596 119885

minus1

119903119905(120596 119909)) = 119909 the implicit function theorem implies

that 119885minus1119903119905(119909) satisfies the first two properties and that

nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)

It follows by Holder inequality and (112) that10038171003817100381710038171003817119863119885

minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment

The author would like to thank the anonymous referees fortheir thorough reviews comments and suggestions signif-icantly contributed to improve the quality of this contribu-tion

References

[1] M Zahle ldquoIntegration with respect to fractal functions andstochastic calculus Irdquo ProbabilityTheory and Related Fields vol111 no 3 pp 333ndash374 1998

[2] D Feyel and A de La Pradelle ldquoOn fractional Brownianprocessesrdquo Potential Analysis vol 10 no 3 pp 273ndash288 1999

[3] F Russo and PVallois ldquoThe generalized covariation process andIto formulardquo Stochastic Processes and their Applications vol 59no 1 pp 81ndash104 1995

[4] F Russo and P Vallois ldquoIto formula for 1198621-functions ofsemimartingalesrdquo Probability Theory and Related Fields vol104 no 1 pp 27ndash41 1996

[5] T J Lyons ldquoDifferential equations driven by rough signalsrdquoRevista Matematica Iberoamericana vol 14 no 2 pp 215ndash3101998

[6] L Coutin and Z Qian ldquoStochastic analysis rough path analysisand fractional Brownian motionsrdquo Probability Theory andRelated Fields vol 122 no 1 pp 108ndash140 2002

[7] L Coutin P Friz and N Victoir ldquoGood rough path sequencesand applications to anticipating stochastic calculusrdquoTheAnnalsof Probability vol 35 no 3 pp 1172ndash1193 2007

[8] L Decreusefond andDNualart ldquoFlow properties of differentialequations driven by fractional Brownian motionrdquo in StochasticDifferential Equations Theory and Applications vol 2 pp 249ndash262 2007

[9] P Friz and N Victoir ldquoApproximations of the Brownianrough path with applications to stochastic analysisrdquo Annales delrsquoInstitut Henri Poincare Probabilites et Statistiques vol 41 no4 pp 703ndash724 2005

[10] P Friz and N Victoir ldquoA note on the notion of geometric roughpathsrdquo Probability Theory and Related Fields vol 136 no 3 pp395ndash416 2006

[11] M Gradinaru I Nourdin F Russo and P Vallois ldquo119898-orderintegrals and generalized Itorsquos formula the case of a fractionalBrownian motion with any Hurst indexrdquo Annales de lrsquoInstitutHenri Poincare Probabilites et Statistiques vol 41 no 4 pp 781ndash806 2005

[12] A Lejay and N Victoir ldquoOn (119901 119902)-rough pathsrdquo Journal ofDifferential Equations vol 225 no 1 pp 103ndash133 2006

[13] T Lyons and N Victoir ldquoAn extension theorem to rough pathsrdquoAnnales de lrsquoInstitut Henri Poincare Analyse Non Lineaire vol24 no 5 pp 835ndash847 2007

[14] A Neuenkirch and I Nourdin ldquoExact rate of convergence ofsome approximation schemes associated to SDEs driven by afractional Brownian motionrdquo Journal of Theoretical Probabilityvol 20 no 4 pp 871ndash899 2007

[15] I Nourdin ldquoSchemas drsquoapproximation associes a une equationdifferentielle dirigee par une fonction holderienne casdu mouvement brownien fractionnairerdquo Comptes RendusMathematique Academie des Sciences Paris vol 340 no 8 pp611ndash614 2005

[16] I Nourdin and T Simon ldquoOn the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motionrdquoStatistics amp Probability Letters vol 76 no 9 pp 907ndash912 2006


[17] I Nourdin and C A Tudor ldquoSome linear fractional stochasticequationsrdquo Stochastics vol 78 no 2 pp 51ndash65 2006

[18] D Feyel and A de La Pradelle ldquoCurvilinear integrals alongenriched pathsrdquo Electronic Journal of Probability vol 11 no 34pp 860ndash892 2006

[19] M Gubinelli ldquoControlling rough pathsrdquo Journal of FunctionalAnalysis vol 216 no 1 pp 86ndash140 2004

[20] J Unterberger ldquoA rough path over multidimensional fractionalBrownianmotion with arbitrary Hurst index by Fourier normalorderingrdquo Stochastic Processes and their Applications vol 120no 8 pp 1444ndash1472 2010

[21] A Neuenkirch S Tindel and J Unterberger ldquoDiscretizing thefractional Levy areardquo Stochastic Processes and their Applicationsvol 120 no 2 pp 223ndash254 2010

[22] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to fractional Brownian motion with Hurst parameterlesser than 12rdquo Stochastic Processes and their Applications vol86 no 1 pp 121ndash139 2000

[23] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to Gaussian processesrdquo The Annals of Probability vol29 no 2 pp 766ndash801 2001

[24] L Decreusefond ldquoStochastic integration with respect to Gaus-sian processesrdquo Comptes Rendus Mathematique Academie desSciences Paris vol 334 no 10 pp 903ndash908 2002

[25] L Decreusefond ldquoStochastic integration with respect to frac-tional Brownian motionrdquo in Theory and Applications of Long-Range Dependence pp 203ndash226 Birkhauser Boston MassUSA 2003

[26] L Decreusefond ldquoStochastic integration with respect toVolterra processesrdquo Annales de lrsquoInstitut Henri Poincare Prob-abilites et Statistiques vol 41 no 2 pp 123ndash149 2005

[27] L Decreusefond and A S Ustunel ldquoStochastic analysis of thefractional Brownian motionrdquo Potential Analysis vol 10 no 2pp 177ndash214 1999

[28] Y Hu and D Nualart ldquoDifferential equations driven by Holdercontinuous functions of order greater than 12rdquo in StochasticAnalysis and Applications vol 2 of The Abel Symposium pp399ndash413 Springer Berlin Germany 2007

[29] D Nualart and A Rascanu ldquoDifferential equations driven byfractional Brownian motionrdquo Universitat de Barcelona Col-lectanea Mathematica vol 53 no 1 pp 55ndash81 2002

[30] D Nualart and B Saussereau ldquoMalliavin calculus for stochasticdifferential equations driven by a fractional Brownian motionrdquoStochastic Processes and their Applications vol 119 no 2 pp 391ndash409 2009

[31] SDarses andB Saussereau ldquoTime reversal for drifted fractionalBrownian motion with Hurst index 1198671199093119890 12rdquo ElectronicJournal of Probability vol 12 no 43 pp 1181ndash1211 2007

[32] P K Friz and N B Victoir Multidimensional Stochastic Pro-cesses as Rough Paths Cambridge University Press CambridgeUK 2010

[33] A S Ustunel and M Zakai ldquoThe construction of filtrations onabstract Wiener spacerdquo Journal of Functional Analysis vol 143no 1 pp 10ndash32 1997

[34] L M Wu ldquoUn traitement unifie de la representation desfonctionnelles de Wienerrdquo in Seminaire de Probabilites XXIVvol 1426 of Lecture Notes in Mathematics pp 166ndash187 SpringerBerlin Germany 1990

[35] M Zakai and O Zeitouni ldquoWhen does the Ramer formula looklike the Girsanov formulardquo The Annals of Probability vol 20no 3 pp 1436ndash1441 1992

[36] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Gordon and Breach Science YverdonSwitzerland 1993

[37] A S Ustunel An Introduction to Analysis on Wiener Space vol1610 ofLectureNotes inMathematics Springer Berlin Germany1995

[38] D NualartTheMalliavin Calculus and Related Topics Springer1995

[39] N Dunford and J T Schwartz Linear Operators Part II WileyClassics Library 1988

[40] A Feintuch and R Saeks System Theory vol 102 of Pure andApplied Mathematics Academic Press New York NY USA1982 A Hilbert Space Approach

[41] A F Nikiforov andV B Uvarov Special Functions ofMathemat-ical Physics Birkhauser 1988

[42] B Simon Trace Ideals and their Applications vol 120 AmericanMathematical Society Providence RI USA 2nd edition 2005

[43] A Uppman ldquoSur le flot drsquoune equation differentielle stochas-tiquerdquo in Seminar on Probability XVI vol 920 pp 268ndash284Springer Berlin Germany 1982

[44] F Hirsch ldquoPropriete drsquoabsolue continuite pour les equationsdifferentielles stochastiques dependant du passerdquo Journal ofFunctional Analysis vol 76 no 1 pp 193ndash216 1988

[45] H Sugita ldquoOn a characterization of the Sobolev spaces overan abstract Wiener spacerdquo Journal of Mathematics of KyotoUniversity vol 25 no 4 pp 717ndash725 1985

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the convergence of which is proved after an integration byparts in the sense of Malliavin calculus Unfortunately thismanipulation generates nonadaptiveness formally the resultcan be expressed as

int

119905

0

119906 (119904) ∘ d119861119867 (119904) = 120575 (Klowast

119905119906) + trace (Klowast

119905nabla119906) (2)

whereK is defined by

K119891 (119905) =119889

119889119905int

119905

0

119870119867 (119905 119904) 119891 (119904) d119904 (3)

and Klowast

119905is the adjoint of K in L2

([0 119905]R) In particularthere exists 119896 such that

Klowast

119905119891 (119904) = int

119905

119904

119896 (119905 119906) 119891 (119906) d119906 (4)

for any 119891 isin L2([0 119905]R) so that even if 119906 is adapted (with

respect to the Brownian filtration) the process (119904 997891rarrKlowast

119905119906(119904))

is anticipative However the stochastic integral process (119905 997891rarrint119905

0119906(119904) ∘ d119861119867(119904)) remains adapted hence the anticipative

aspect is in some sense artificialThemotivation of this workis to show that up to time reversal we can work with adaptedprocess and Ito integralsThe time-reversal properties of fBmwere already studied in [31] in a different context It wasshown there that the time reversal of the solution of an fBm-driven SDE of the form

119889119884 (119905) = 119906 (119884 (119905)) d119905 + d119861119867 (119905) (5)

is still a process of the same form With a slight adaptationof our method to fBm-driven SDEs with drift one shouldrecover the main theorem of [31]

In what follows there is no restriction on the dimensionbut we need to assume that any component of 119861119867 is an fBm ofHurst index greater than 12 Consider that we want to solvethe following equation

119883119905= 119909 + int

119905

0

120590 (119883119904) ∘ d119861119867 (119904) 0 le 119905 le 119879 (6)

where 120590 is a deterministic function whose properties will befixed below It turns out that it is essential to investigate themore general equations

119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119861119867 (119904) 0 le 119903 le 119905 le 119879 (A1015840

)

The strategy is then as follows We will first consider thereciprocal problem

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119861119867 (119904) 0 le 119903 le 119905 le 119879 (B1015840)

The first critical point is that when we consider 119885119903119905=

119884119905minus119903119905

119903 isin [0 119905] this process solves an adapted past-dependent and stochastic differential equation with respectto a standard Brownian motion Moreover because 119870

119867

is lower-triangular and sufficiently regular the trace term

vanishes in the equation defining 119885 We have then reducedthe problem to an SDE with coefficients dependent on thepast a problem which can be handled by the usual contrac-tion methods We do not claim that the results presentedare new (for instance see the brilliant monograph [32] fordetailed results obtained via rough paths theory) but it seemsinteresting to have purely probabilistic methods which showthat fBm driven SDEs do have strong solutions which arehomeomorphisms Moreover the approach given here showsthe irreducible difference between the case119867 lt 12 and119867 gt

12 The trace term only vanishes in the latter situation sothat such an SDE is merely a usual SDE with past-dependentcoefficients This representation may be fruitful for instanceto analyze the support and prove the absolute continuity ofsolutions of (6)

This paper is organized as follows After some preliminar-ies on fractional Sobolev spaces often called Besov-Liouvillespace we address in Section 3 the problem of Malliavincalculus and time reversal This part is interesting in its ownsince stochastic calculus of variations is a framework oblivi-ous to time Constructing such a notion of time is achievedusing the notion of resolution of the identity as introduced in[33 34] We then introduce the second key ingredient whichis the notion of strict causality or quasinilpotence see [35]for a related application In Section 4 we show that solving(B1015840) reduces to solve a past-dependent stochastic differentialequation with respect to a standard Brownian motion see(C) below Then we prove existence uniqueness and someproperties of this equation Technical lemmas are postponedto Section 5

2 Besov-Liouville Spaces

Let 119879 gt 0 be fix real number For a measurable function 119891 [0 119879] rarr R119899 we define 120591

119879119891 by

120591119879119891 (119904) = 119891 (119879 minus 119904) for any 119904 isin [0 119879] (7)

For 119905 isin [0 119879] 119890119905119891 will represent the restriction of 119891 to [0 119905]

that is 119890119905119891 = 1198911

[0119905] For any linear map 119860 we denote by 119860lowast

119879

its adjoint in L2([0 119879]R119899) For 120578 isin (0 1] the space of 120578-

Holder continuous functions on [0 119879] is equipped with thenorm

10038171003817100381710038171198911003817100381710038171003817Hol(120578) = sup

0lt119904lt119905lt119879

1003816100381610038161003816119891 (119905) minus 119891 (119904)1003816100381610038161003816

|119905 minus 119904|120578

+10038171003817100381710038171198911003817100381710038171003817infin (8)

Its topological dual is denoted by Hol(120578)lowast For 119891 isin

L1([0 119879] R119899 d119905) (denoted by L1 for short) the left and

right fractional integrals of 119891 are defined by

(119868120574

0+119891) (119909) =

1

Γ (120574)int

119909

0

119891 (119905) (119909 minus 119905)120574minus1d119905 119909 ge 0

(119868120574

119879minus119891) (119909) =

1

Γ (120574)int

119879

119909

119891 (119905) (119905 minus 119909)120574minus1d119905 119909 le 119879

(9)


where 120574 gt 0 and 11986800+ = 119868

0



int

119879

0

119891 (119904) (119868120574

0+119892) (119904) d119904 = int

119879

0

(119868120574

119879minus119891) (119904) 119892 (119904) d119904 (10)


) = I+



0+11989110038171003817100381710038171003817I+120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(11)


) = Iminus

120574119901


119879minus11989110038171003817100381710038171003817Iminus120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(12)



0+is a bounded













1

0+(L


([0 119879])




119889

119894=1119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩)119909

119894 where for each 119894119891

119894isin

Cinfin

0(R119899R) and (119907



nablaW120601 (119908) = sum

119894119895=1

120597119895119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩) 119907

119894119895otimes 119909

119894

(13)





10038171003817100381710038171206011003817100381710038171003817119901119896

=

119896

sum

119894=0

10038171003817100381710038171003817(nabla


(14)


the space D1199011(L119901



100381610038161003816100381610038161003816100381610038161003816

E [int119879

0

119907119904nablaW119904120601d119904]

100381610038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171206011003817100381710038171003817119871119901

(15)


E [int119879

0

119907119904nablaW119904120601d119904] = E [120601 120575W119907] (16)


Θ119879 Ω 997888rarr Ω

120596 997891997888rarr = 120596 (119879) minus 120591119879120596

(17)


ℒ2

ℒ2

120591119879


1198681

0+ 119868

1

0+

(18)

Note that Θminus1

119879= Θ


Cinfin


nabla119903119891 (120596 (119905

1) 120596 (119905

119896))

=

119896

sum

119895=1

120597119895119891 (120596 (119905

1) 120596 (119905

119896)) 1

[0119905119895] (119903)

nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[0119905119895] (119903)

(19)



119891 ( (1199051) (119905

119896))


(20)


we can consider 119891((1199051) (119905



nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[119879minus119905119895119879] (119903)

(21)


nabla119865 ∘ Θ119879 (120596) = 120591119879nabla119865 () (22)

Since Θlowast

119879P = P and 120591



and D119901119896



and D119901119896


for some 119901 119896

nabla (119865 ∘ Θ119879) = 120591

119879nabla (119865 ∘ Θ

119879) P as (23)




equality holds

120575 (119906 ()) () = 120575 (120591119879119906 ()) (120596) = 120575 (120591119879119906 ∘ Θ119879) (120596) (24)


E [119865 () 120575119906 ()] = E [(nabla119865 () 119906)L2] (25)


E [(nabla119865 () 119906)L2] = E [(120591

119879nabla119865 ∘ Θ

119879 (120596) 119906)L2]

= E [(nabla119865 ∘ Θ119879 (120596) 120591119879119906)L2

]

= E [119865 ∘ Θ119879 (120596) 120575 (120591119879119906) (120596)]

= E [119865 () 120575 (120591119879119906) (120596)]

(26)


120575119906 = sum

119894

((119906 ℎ119894)L2120575ℎ


119894otimes ℎ

119894)L2otimesL2

) (27)


([0 119879] R119899)Thus we have

120575 (119906 ()) () = sum

119894

((119906 () ℎ119894)L2120575ℎ119894 ()


= sum

119894

((119906 () ℎ119894)L2120575 (120591

119879ℎ119894) (120596)


= sum

119894

((120591119879119906 () 120591119879ℎ119894)L2

120575 (120591119879ℎ119894) (120596)


)

(28)







119899 119899 ge 1) ofL2

([0 119879] R119899)

sum

119899ge1

1003816100381610038161003816(119881ℎ119899 ℎ119899)L21003816100381610038161003816 is finite (29)


trace (119881) = sum119899ge1

(119881ℎ119899 ℎ

119899)L2 (30)





(1) 1198640= 0 and 119864

1= Id

(2) 119864120582119864120583= 119864

120582and120583

(3) lim120583darr120582119864120583= 119864


120583uarr1119864120583= Id






|120587| = sup119894|119905119894+1minus 119905

119894|





119864120582119881119864

120582= 119864

120582119881 for any 120582 isin [0 1] (31)


int119879




0lt 119905


119899= 119879 sub 120587

10038171003817100381710038171003817(119864

119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)10038171003817100381710038171003817L2

lt 120576 for 119894 = 0 119899 minus 1(32)


1003817100381710038171003817(119864119905 minus 119864119904)119881(119864119905 minus 119864119904)1003817100381710038171003817L2

=1003817100381710038171003817119864119905 minus 119864119904

1003817100381710038171003817L2= 1 (33)

since 119864119905minus 119864




120582119879 120582 isin [0 1]) or 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) If 119881 is





1003817100381710038171003817(119864119905 minus 119864119904) 119881 (119864119905 minus 119864119904) 1198911003817100381710038171003817L2

= int

119905

119904

1003816100381610038161003816119881(1198911(119904119905])(119906)1003816100381610038161003816

2d119906

le (119905 minus 119904)1minus21199011003817100381710038171003817119881(1198911(119904119905])

1003817100381710038171003817L1199012

le 119888 (119905 minus 119904)1minus21199011003817100381710038171003817119891

1003817100381710038171003817L2

(34)


implies (119864119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894


119899= 119879 sub 120587 and any 119894 = 0 119899 minus 1






F119864

119905= 120590 120575

W(119864

120582ℎ) 120582 le 119905 ℎ isinL

2 (35)


120582119906 ℎ⟩ is F119864-














21(L2





Theorem 10


120582 119879119891 = 1198911

[0120582119879]and 0 is


[(1minus120582)119879119879]


and F0






119879119881lowast

119879120591119879is 0-causal


120591119879119890119903= (Id minus 119890

119879minus119903) 120591

119879for any 0 le 119903 le 119879 (36)


120591119879119890119903119891 (119904) = 119891 (119879 minus 119904) 1[0119903] (119879 minus 119904)

= 119891 (119879 minus 119904) 1[119879minus119903119879] (119904)

= (Id minus 119890119879minus119903) 120591

119879119891 (119904) for any 0 le 119904 le 119879

(37)



119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890


119879= 120591


119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)


119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)






119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)




sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)


0119906(119904) ∘ d119882119881

(119904)


1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)


119879minus



int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1


1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)



+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)


(47)



([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)


minus1


int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)


119867)lowast







119906


int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)


E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)



int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)



E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120574 gt 0 and 11986800+ = 119868

0



int

119879

0

119891 (119904) (119868120574

0+119892) (119904) d119904 = int

119879

0

(119868120574

119879minus119891) (119904) 119892 (119904) d119904 (10)


) = I+



0+11989110038171003817100381710038171003817I+120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(11)


) = Iminus

120574119901


119879minus11989110038171003817100381710038171003817Iminus120574119901

=10038171003817100381710038171198911003817100381710038171003817L119901

(12)



0+is a bounded













1

0+(L


([0 119879])




119889

119894=1119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩)119909

119894 where for each 119894119891

119894isin

Cinfin

0(R119899R) and (119907



nablaW120601 (119908) = sum

119894119895=1

120597119895119891119894(⟨119907

1198941 119908⟩ ⟨119907

119894119899 119908⟩) 119907

119894119895otimes 119909

119894

(13)





10038171003817100381710038171206011003817100381710038171003817119901119896

=

119896

sum

119894=0

10038171003817100381710038171003817(nabla


(14)


the space D1199011(L119901



100381610038161003816100381610038161003816100381610038161003816

E [int119879

0

119907119904nablaW119904120601d119904]

100381610038161003816100381610038161003816100381610038161003816

le 11988810038171003817100381710038171206011003817100381710038171003817119871119901

(15)


E [int119879

0

119907119904nablaW119904120601d119904] = E [120601 120575W119907] (16)


Θ119879 Ω 997888rarr Ω

120596 997891997888rarr = 120596 (119879) minus 120591119879120596

(17)


ℒ2

ℒ2

120591119879


1198681

0+ 119868

1

0+

(18)

Note that Θminus1

119879= Θ


Cinfin


nabla119903119891 (120596 (119905

1) 120596 (119905

119896))

=

119896

sum

119895=1

120597119895119891 (120596 (119905

1) 120596 (119905

119896)) 1

[0119905119895] (119903)

nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[0119905119895] (119903)

(19)



119891 ( (1199051) (119905

119896))


(20)


we can consider 119891((1199051) (119905



nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[119879minus119905119895119879] (119903)

(21)


nabla119865 ∘ Θ119879 (120596) = 120591119879nabla119865 () (22)

Since Θlowast

119879P = P and 120591



and D119901119896



and D119901119896


for some 119901 119896

nabla (119865 ∘ Θ119879) = 120591

119879nabla (119865 ∘ Θ

119879) P as (23)




equality holds

120575 (119906 ()) () = 120575 (120591119879119906 ()) (120596) = 120575 (120591119879119906 ∘ Θ119879) (120596) (24)


E [119865 () 120575119906 ()] = E [(nabla119865 () 119906)L2] (25)


E [(nabla119865 () 119906)L2] = E [(120591

119879nabla119865 ∘ Θ

119879 (120596) 119906)L2]

= E [(nabla119865 ∘ Θ119879 (120596) 120591119879119906)L2

]

= E [119865 ∘ Θ119879 (120596) 120575 (120591119879119906) (120596)]

= E [119865 () 120575 (120591119879119906) (120596)]

(26)


120575119906 = sum

119894

((119906 ℎ119894)L2120575ℎ


119894otimes ℎ

119894)L2otimesL2

) (27)


([0 119879] R119899)Thus we have

120575 (119906 ()) () = sum

119894

((119906 () ℎ119894)L2120575ℎ119894 ()


= sum

119894

((119906 () ℎ119894)L2120575 (120591

119879ℎ119894) (120596)


= sum

119894

((120591119879119906 () 120591119879ℎ119894)L2

120575 (120591119879ℎ119894) (120596)


)

(28)







119899 119899 ge 1) ofL2

([0 119879] R119899)

sum

119899ge1

1003816100381610038161003816(119881ℎ119899 ℎ119899)L21003816100381610038161003816 is finite (29)


trace (119881) = sum119899ge1

(119881ℎ119899 ℎ

119899)L2 (30)





(1) 1198640= 0 and 119864

1= Id

(2) 119864120582119864120583= 119864

120582and120583

(3) lim120583darr120582119864120583= 119864


120583uarr1119864120583= Id






|120587| = sup119894|119905119894+1minus 119905

119894|





119864120582119881119864

120582= 119864

120582119881 for any 120582 isin [0 1] (31)


int119879




0lt 119905


119899= 119879 sub 120587

10038171003817100381710038171003817(119864

119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)10038171003817100381710038171003817L2

lt 120576 for 119894 = 0 119899 minus 1(32)


1003817100381710038171003817(119864119905 minus 119864119904)119881(119864119905 minus 119864119904)1003817100381710038171003817L2

=1003817100381710038171003817119864119905 minus 119864119904

1003817100381710038171003817L2= 1 (33)

since 119864119905minus 119864




120582119879 120582 isin [0 1]) or 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) If 119881 is





1003817100381710038171003817(119864119905 minus 119864119904) 119881 (119864119905 minus 119864119904) 1198911003817100381710038171003817L2

= int

119905

119904

1003816100381610038161003816119881(1198911(119904119905])(119906)1003816100381610038161003816

2d119906

le (119905 minus 119904)1minus21199011003817100381710038171003817119881(1198911(119904119905])

1003817100381710038171003817L1199012

le 119888 (119905 minus 119904)1minus21199011003817100381710038171003817119891

1003817100381710038171003817L2

(34)


implies (119864119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894


119899= 119879 sub 120587 and any 119894 = 0 119899 minus 1






F119864

119905= 120590 120575

W(119864

120582ℎ) 120582 le 119905 ℎ isinL

2 (35)


120582119906 ℎ⟩ is F119864-














21(L2





Theorem 10


120582 119879119891 = 1198911

[0120582119879]and 0 is


[(1minus120582)119879119879]


and F0






119879119881lowast

119879120591119879is 0-causal


120591119879119890119903= (Id minus 119890

119879minus119903) 120591

119879for any 0 le 119903 le 119879 (36)


120591119879119890119903119891 (119904) = 119891 (119879 minus 119904) 1[0119903] (119879 minus 119904)

= 119891 (119879 minus 119904) 1[119879minus119903119879] (119904)

= (Id minus 119890119879minus119903) 120591

119879119891 (119904) for any 0 le 119904 le 119879

(37)



119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890


119879= 120591


119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)


119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)






119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)




sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)


0119906(119904) ∘ d119882119881

(119904)


1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)


119879minus



int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1


1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)



+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)


(47)



([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)


minus1


int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)


119867)lowast







119906


int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)


E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)



int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)



E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










we can consider 119891((1199051) (119905



nabla119903119891 ( (119905

1) (119905

119896))

=

119896

sum

119895=1

120597119895119891 ( (119905

1) (119905

119896)) 1

[119879minus119905119895119879] (119903)

(21)


nabla119865 ∘ Θ119879 (120596) = 120591119879nabla119865 () (22)

Since Θlowast

119879P = P and 120591



and D119901119896



and D119901119896


for some 119901 119896

nabla (119865 ∘ Θ119879) = 120591

119879nabla (119865 ∘ Θ

119879) P as (23)




equality holds

120575 (119906 ()) () = 120575 (120591119879119906 ()) (120596) = 120575 (120591119879119906 ∘ Θ119879) (120596) (24)


E [119865 () 120575119906 ()] = E [(nabla119865 () 119906)L2] (25)


E [(nabla119865 () 119906)L2] = E [(120591

119879nabla119865 ∘ Θ

119879 (120596) 119906)L2]

= E [(nabla119865 ∘ Θ119879 (120596) 120591119879119906)L2

]

= E [119865 ∘ Θ119879 (120596) 120575 (120591119879119906) (120596)]

= E [119865 () 120575 (120591119879119906) (120596)]

(26)


120575119906 = sum

119894

((119906 ℎ119894)L2120575ℎ


119894otimes ℎ

119894)L2otimesL2

) (27)


([0 119879] R119899)Thus we have

120575 (119906 ()) () = sum

119894

((119906 () ℎ119894)L2120575ℎ119894 ()


= sum

119894

((119906 () ℎ119894)L2120575 (120591

119879ℎ119894) (120596)


= sum

119894

((120591119879119906 () 120591119879ℎ119894)L2

120575 (120591119879ℎ119894) (120596)


)

(28)







119899 119899 ge 1) ofL2

([0 119879] R119899)

sum

119899ge1

1003816100381610038161003816(119881ℎ119899 ℎ119899)L21003816100381610038161003816 is finite (29)


trace (119881) = sum119899ge1

(119881ℎ119899 ℎ

119899)L2 (30)





(1) 1198640= 0 and 119864

1= Id

(2) 119864120582119864120583= 119864

120582and120583

(3) lim120583darr120582119864120583= 119864


120583uarr1119864120583= Id






|120587| = sup119894|119905119894+1minus 119905

119894|





119864120582119881119864

120582= 119864

120582119881 for any 120582 isin [0 1] (31)


int119879




0lt 119905


119899= 119879 sub 120587

10038171003817100381710038171003817(119864

119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)10038171003817100381710038171003817L2

lt 120576 for 119894 = 0 119899 minus 1(32)


1003817100381710038171003817(119864119905 minus 119864119904)119881(119864119905 minus 119864119904)1003817100381710038171003817L2

=1003817100381710038171003817119864119905 minus 119864119904

1003817100381710038171003817L2= 1 (33)

since 119864119905minus 119864




120582119879 120582 isin [0 1]) or 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) If 119881 is





1003817100381710038171003817(119864119905 minus 119864119904) 119881 (119864119905 minus 119864119904) 1198911003817100381710038171003817L2

= int

119905

119904

1003816100381610038161003816119881(1198911(119904119905])(119906)1003816100381610038161003816

2d119906

le (119905 minus 119904)1minus21199011003817100381710038171003817119881(1198911(119904119905])

1003817100381710038171003817L1199012

le 119888 (119905 minus 119904)1minus21199011003817100381710038171003817119891

1003817100381710038171003817L2

(34)


implies (119864119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894


119899= 119879 sub 120587 and any 119894 = 0 119899 minus 1






F119864

119905= 120590 120575

W(119864

120582ℎ) 120582 le 119905 ℎ isinL

2 (35)


120582119906 ℎ⟩ is F119864-














21(L2





Theorem 10


120582 119879119891 = 1198911

[0120582119879]and 0 is


[(1minus120582)119879119879]


and F0






119879119881lowast

119879120591119879is 0-causal


120591119879119890119903= (Id minus 119890

119879minus119903) 120591

119879for any 0 le 119903 le 119879 (36)


120591119879119890119903119891 (119904) = 119891 (119879 minus 119904) 1[0119903] (119879 minus 119904)

= 119891 (119879 minus 119904) 1[119879minus119903119879] (119904)

= (Id minus 119890119879minus119903) 120591

119879119891 (119904) for any 0 le 119904 le 119879

(37)



119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890


119879= 120591


119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)


119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)






119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)




sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)


0119906(119904) ∘ d119882119881

(119904)


1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)


119879minus



int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1


1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)



+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)


(47)



([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)


minus1


int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)


119867)lowast







119906


int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)


E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)



int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)



E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864120582119881119864

120582= 119864

120582119881 for any 120582 isin [0 1] (31)


int119879




0lt 119905


119899= 119879 sub 120587

10038171003817100381710038171003817(119864

119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894

)10038171003817100381710038171003817L2

lt 120576 for 119894 = 0 119899 minus 1(32)


1003817100381710038171003817(119864119905 minus 119864119904)119881(119864119905 minus 119864119904)1003817100381710038171003817L2

=1003817100381710038171003817119864119905 minus 119864119904

1003817100381710038171003817L2= 1 (33)

since 119864119905minus 119864




120582119879 120582 isin [0 1]) or 119864 = (Id minus 119890

(1minus120582)119879 120582 isin [0 1]) If 119881 is





1003817100381710038171003817(119864119905 minus 119864119904) 119881 (119864119905 minus 119864119904) 1198911003817100381710038171003817L2

= int

119905

119904

1003816100381610038161003816119881(1198911(119904119905])(119906)1003816100381610038161003816

2d119906

le (119905 minus 119904)1minus21199011003817100381710038171003817119881(1198911(119904119905])

1003817100381710038171003817L1199012

le 119888 (119905 minus 119904)1minus21199011003817100381710038171003817119891

1003817100381710038171003817L2

(34)


implies (119864119905119894+1

minus 119864119905119894

)119881(119864119905119894+1

minus 119864119905119894


119899= 119879 sub 120587 and any 119894 = 0 119899 minus 1






F119864

119905= 120590 120575

W(119864

120582ℎ) 120582 le 119905 ℎ isinL

2 (35)


120582119906 ℎ⟩ is F119864-














21(L2





Theorem 10


120582 119879119891 = 1198911

[0120582119879]and 0 is


[(1minus120582)119879119879]


and F0






119879119881lowast

119879120591119879is 0-causal


120591119879119890119903= (Id minus 119890

119879minus119903) 120591

119879for any 0 le 119903 le 119879 (36)


120591119879119890119903119891 (119904) = 119891 (119879 minus 119904) 1[0119903] (119879 minus 119904)

= 119891 (119879 minus 119904) 1[119879minus119903119879] (119904)

= (Id minus 119890119879minus119903) 120591

119879119891 (119904) for any 0 le 119904 le 119879

(37)



119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890


119879= 120591


119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)


119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)






119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)




sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)


0119906(119904) ∘ d119882119881

(119904)


1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)


119879minus



int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1


1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)



+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)


(47)



([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)


minus1


int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)


119867)lowast







119906


int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)


E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)



int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)



E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890119903120591119879119881lowast

119879120591119879119890119903= 119890

119903120591119879119881lowast

119879120591119879or equivalently (38)

119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903 (39)

since 119890lowast119903= 119890


119879= 120591


119890119903120591119879119881120591

119879119890119903= 120591

119879119881120591

119879119890119903minus 119890

119879minus119903119881120591

119879119890119903 (40)


119890119879minus119903119881120591

119879119890119903= 119890

119879minus119903119881 (Id minus 119890

119879minus119903) 120591

119879

= (119890119879minus119903119881 minus 119890

119879minus119903119881119890

119879minus119903) 120591

119879= 0

(41)






119882119881(119905) = 120575

W(119881 (1

[0119905])) forall 119905 isin [0 119879] (42)




sums

119877120587(119905 119906) = 120575

W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881119906 (119903) d119903 1[119905119894119905119894+1))

+ sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119906) (119904) d119904 d119903

(43)


0119906(119904) ∘ d119882119881

(119904)


1

Γ (119867 + 12)int

119905

0

(119905 minus 119904)119867minus12d119861

119904= 120575 (119868

119867minus12

119879minus (1

[0119905])) (44)


119879minus



int

119905

0

119870119867 (119905 119904) d119861 (119904) (45)

where

119870119867 (119905 119903)

=(119905 minus 119903)

119867minus(12)

Γ (119867 + (12))119865 (

1


1

2119867 +

1

2 1 minus

119905

119903)

times 1[0119905) (119903)

(46)



+infin

sum

119896=0

(120572)119896(120573)119896

(120574)119896119896119911119896

(119886)0 = 1

(119886)119896 =Γ (119886 + 119896)


(47)



([0 1]) ontoI+

119867+12119901and

119870119867119891 = 119868

1

0+119909

119867minus12119868119867minus12

0+ 119909

12minus119867119891 (48)


minus1


int

119905

0

119870119867 (119905 119904) d119861 (119904) = int

119905

0

(K119867)lowast

119879(1[0119905]) (119904) d119861 (119904) (49)


119867)lowast







119906


int

119879

0

119906 (119904) ∘ dWV(s) = 120575W (Vu) + int

T

0DWu (s) ds (50)


E [int119879

0

10038161003816100381610038161003816119863

W119906(119903)

10038161003816100381610038161003816

119901

dr] le 119888 119879119901120578119906119901L1199011

(51)



int

119903

0

119906 (119904) ∘ dWV(s)

= int

119879

0

(119890119903119906) (119904) ∘ dWV

(s)

= 120575W(119881119890

119903119906) + int

119903

0

119863W119906 (119904) ds

(52)



E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E[10038171003817100381710038171003817100381710038171003817

int

0

119906(119904) ∘ dWV(s)10038171003817100381710038171003817100381710038171003817

119901

Hol (120578)] le 119888 119906

119901

L1199011

(53)




1199011and let

119879= 120591

119905119881120591



119879isF

0-adapted Then

int

119879minus119903

119879minus119905

120591119879119906 (119904) ∘ dWV

(s)

= int

119905

119903

119879(1[119903119905]) (119904) dBT



119879(119861)(119904)


119879= 119868

119867minus12

0+


0+



119879

0

int

119904

0

(119904 minus 119903)119867minus32

nabla119904119906 (119903) d119903 d119904 = 0

(55)




119879nabla119906 to be




119889 (120593 120601) = 120588 (120593 120601) + 120588 (120593minus1 120601

minus1) (56)

where

120588 (120593 120601) =

infin

sum

119873=1

2minus119873

sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

1 + sup|119909|le119873

1003816100381610038161003816120593 (119909) minus 120601 (119909)1003816100381610038161003816

sdot (57)


119883119903119905= 119909 + int

119905

119903

120590 (119883119903119904) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (A)

119884119903119905= 119909 minus int

119905

119903

120590 (119884119904119905) ∘ d119882119881

(119904) 0 le 119903 le 119905 le 119879 (B)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119883119903119905 (120596 119909))

(58)



120590119882119881(119904) 119903 le 119904 le 119905-measurable


119903119905(120596 119909) and (120596 119905) 997891rarr 119883

minus1

119903119905(120596 119909) belong to

L1199011



119883119903119905 (120596 119909) = 119883119904119905

(120596119883119903119904 (120596 119909)) (59)




(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119884119903119905 (120596 119909))

(60)



120590119882119881(119904) 119903 le 119904 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119884

minus1

119903119905(120596 119909) belong to

L1199011



119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (61)


119885119903119905= 119909 minus int

119905

119903

119879(120590 ∘ 119885

1199051[119903119905]) (119904) d119879 (119904) (C)





(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(120596 119903 119905) 997891997888rarr (119909 997891997888rarr 119885119903119905 (120596 119909))

(62)



120590119879(119904) 119904 le 119903 le 119905measurable


119903119905(120596 119909) and (120596 119903) 997891rarr 119885

minus1

119903119905(120596 119909) belong to

L1199011



from L119901 into I120572119901


E [1003816100381610038161003816119885119903119879 minus 1198851199031015840 1198791003816100381610038161003816

119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(63)

Moreover

(120596 119903) 997891997888rarr 119885119903119904(120596 119885

119904119905 (120596 119909)) isin L1199011

for any 119903 le 119904 le 119905 le 119879(64)



fromL119901 intoI120572119901


Proof Set

119885119903119879 ( 119909) = 119884119879minus119903119879 (Θ

minus1

119879() 119909) (65)

or equivalently

119884119903119879 (120596 119909) = 119885119879minus119903119879 (Θ119879 (120596) 119909) (66)


119879



uous from L119901 into I120572119901


119884119903119905 (120596 119909) = 119884119903119904 (120596 119884119904119905 (120596 119909)) (67)



119904119905(120596 119909))


and119884119903119904(120596 119884

119904119905 (120596 119909))

= 119884119904119905 (120596 119909) minus int

119904

119903

120590 (119884120591119904 (120596 119909)) ∘ d119882

119881(120591)

10038161003816100381610038161003816100381610038161003816119909=119884119904119905(120596119909)

= 119909 minus int

119905

119904

120590 (119884120591119905 (120596 119909) ∘ d119882

119881(120591)

minus int

119904

119903

120590 (119884120591119904(120596 119884

119904119905 (120596 119909)) ) ∘ d119882119881(120591)

(68)

Set

119877120591119905=

119884120591119905 (120596 119909) for 119904 le 120591 le 119905119884120591119904(120596 119884

119904119905 (120596 119909)) for 119903 le 120591 le 119904(69)


119904119905(120596 119909) = 119884




E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(70)



remark that 1198841199041199041015840(119909)


E [1003816100381610038161003816119884119903119904(119909) minus 11988411990310158401199041015840(119909)1003816100381610038161003816

119901]

le 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 1199041015840 (119909)1003816100381610038161003816

119901])

= 119888 (E [1003816100381610038161003816119884119903119904 (119909) minus 1198841199031015840 119904 (119909)1003816100381610038161003816

119901]

+E [10038161003816100381610038161198841199031015840 119904 (119909) minus 1198841199031015840 119904 (1198841199041199041015840 (119909))1003816100381610038161003816

119901])


119901]


119901])

(71)



119901] le 119888

10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

(1 + |119909|119901) (72)


119885119904minus1199031015840119904(119884

1199041199041015840 (119909)) = 119885119904minus1199031015840 119904(119910)

1003816100381610038161003816119910=1198841199041199041015840 (119909) (73)




119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901]

le 119888E [1003816100381610038161003816119909 minus 1198841199041199041015840 (119909)1003816100381610038161003816

119901]

(74)


01199041015840 minus 119885

1199041015840minus1199041199041015840(119909)|

119901] thus we get


119901]

le 119888 (1 + |119909|119901)100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

(75)


E [1003816100381610038161003816119884119903119904(119909) minus 1198841199031015840 1199041015840(119909)1003816100381610038161003816

119901]

le 119888 (1 + |119909|119901) (100381610038161003816100381610038161199041015840minus 11990410038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

)

(76)




fromL119901 intoI120572119901



minus1

119904119905(120596 119909) belongs to L

1199011 hence



5 Technical Proofs



1199011 If 119881nablaW


int

119879

0


Moreover

E [10038161003816100381610038161003816 trace(119881nablaW119906)10038161003816100381610038161003816

119901

] le 119888119906119901

L1199011

(78)



120601119896119898

= 211989621



span of the 120601119896119898


[42])

trace (119881nablaW119901119905119906) = lim

119896rarr+infin

trace (119875119896119881nabla

W119901119905119906 119875

119896) (79)

Now

trace (119875119896119881nabla

W119906 119875

119896)

=

119896

sum

119898=1

(119881nablaW119901119905119906 120601

119896119898otimes 120601

119896119898)L2otimesL2

=

119896

sum

119898=1

2119896int

1198982minus119896and119905

(119898minus1)2minus119896and119905

int

1198982minus119896and119905

(119898minus1)2minus119896and119905

119881(nablaW119903119906) (119904) d119904 d119903

(80)



119906 R119898 997888rarr L1199011

119909 997891997888rarr ((120596 119904) 997891997888rarr 119906 (120596 119904 119909))

(81)


119901119870 (119906) = sup

119909isin119870

119906(119909)L1199011 (82)



119901 Let 119865 be a


Then

int

119879

0

119906 (119904 119865) ∘ d119882119881(119904) = int

119879

0

119906(119904 119909) ∘ d119882119881

119904

100381610038161003816100381610038161003816100381610038161003816119909=119865

(83)


119906 (120596 119904 119909) =

119870

sum

119897=1

119867119897 (119909) 119906119897 (120596 119904) (84)

with119867119897smooth and 119906

119897in L


119901 In view of (53)


120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119867(119865)119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

= 119867 (119865) 120575W(sum

119905119894isin120587

1

120579119894

int

119905119894+1and119905

119905119894and119905

119881 (119906 (120596 )) (119903) d119903 1[119905119894119905119894+1))

minus sum

119905119894isin120587

int

119905119894+1and119905

119905119894and119905

int

119905119894+1and119905

119905119894and119905

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(85)

On the other hand

nablaW119904(119867 (119865) 119906 (120596 119903)) = 119867

1015840(119865) nabla

W119904119865 119906 (119903) (86)


Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Hence

sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

119881(nablaW119903119867(119865) 119906) (119904) d119904 d119903

= sum

119905119894isin120587

1

120579119894

∬[119905119894and119905119905119894+1and119905]2

1198671015840(119865) nabla

W119904119865 119881119906 (119903) d119904 d119903

(87)



int119905

119903119906(119904) ∘ d119882119881

(119904) as

int

119905

119903

119906 (119904) ∘ d119882119881(119904)

= int

119905

0

119906 (119904) ∘ d119882119881(119904) minus int

119903

0

119906 (119904) ∘ d119882119881(119904)

= int

119879

0

119890119905119906 (119904) d119882119881

(119904) minus int

119879

0

119890119903119906 (119904) ∘ d119882119881

(119904)

= 120575W(119881 (119890

119905minus 119890

119903) 119906) + int

119905

119903

119863W119906 (119904) d119904

(88)



119879119860otimes119861 (resp119860120591

119879otimes119861)


119879)


trace (120591119879119860 otimes 119861)


119879otimes 119861)

= trace (119860 otimes 119861120591119879)

(89)




119879119881119906 andnablaW

otimes119881120591119879119906





119879119881119906) = trace (120591

119879nablaWotimes 119881119906)


119879) 119906) = trace (nablaW

120591119879otimes 119881119906)

(90)


120575119861(119881 (119890

119879minus119903minus 119890

119879minus119905) 120591

119879 ∘ Θ

119879)

= 120575119861(119881120591

119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575119861(120591119879119879(119890119905minus 119890

119903) ∘ Θ

119879)

= 120575 (119879(119890119905minus 119890

119903) ) ()

= int

119905

119903

119879(1[119903119905]) (119904) d119861119879 (119904)

(91)




that nabla119881(119890119905minus 119890



120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903) (92)


120591119879119879120591119879otimes 120591

119879nabla120591

119879(119890119905minus 119890

119903)

= 119881 (nabla120591119879(119890119879minus119903

minus 119890119879minus119905) ∘ Θ

119879)

(93)




119879∘ 120590 119862 ([0 119879] R119899) 997888rarr 119862 ([0 119879] R119899)

120601 997891997888rarr 119879(120590 ∘ 120595 1

[119886119887])

(94)


119889119879∘ 120590 (120601) [120595] =

119879(120590

1015840∘ 120601 120595) (95)


|120590 (119909)| le 119888 (1 + |119909|) for any 119909 isin R119899 (96)


10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888119879

120578+1119901(1 + int

119905

0

1003816100381610038161003816120595 (119904)1003816100381610038161003816

119901ds)

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(97)


119879(120590 ∘120595minus


sup119905le119879

10038161003816100381610038161003816119879(120590 ∘ 1205951

[119886119887]) (119905) minus 119879 (120590 ∘ 120601 1

[119886119887]) (119905)

10038161003816100381610038161003816

le 11988810038171003817100381710038171003817119879((120590 ∘ 120595 minus 120590 ∘ 120601) 1

[119886119887])10038171003817100381710038171003817Hol(120578)

le 1198881003817100381710038171003817(120590 ∘ 120595 minus 120590 ∘ 120601) 1[119886119887]

1003817100381710038171003817L119901

le 1198881003817100381710038171003817120601 minus 120595

1003817100381710038171003817L119901([119886119887])

le 119888 sup119905le119879

1003816100381610038161003816120595 (119905) minus 120601 (119905)1003816100381610038161003816

(98)



120590 (120595 (119905) + 120576120601 (119905))

= 120590 (120595 (119905)) + 120576int

1

0

1205901015840(119906120595 (119905) + (1 minus 119906) 120601 (119905)) d119906

(99)



int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

119879

0

100381610038161003816100381610038161003816100381610038161003816

int

1

0

1205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906

100381610038161003816100381610038161003816100381610038161003816

119901

d119905 le 119888119879 (100)


01205901015840(119906120595(119905) + (1 minus 119906)120601(119905))d119906) belongs


119879(int

1

0

1205901015840(119906120595() + (1 minus 119906)120601()) d)

100381710038171003817100381710038171003817100381710038171003817119862

le 119888119879 (101)

Thus

lim120576rarr0

120576minus1(

119879(120590 ∘ (120595 + 120576120601)) minus

119879(120590 ∘ 120595)) exists (102)



esis 1 we have

10038161003816100381610038161003816119879(120590 ∘ 120595) (119905)

10038161003816100381610038161003816le 119888(int

119905

0

1199041205781199011003816100381610038161003816120590(120595(119904))

1003816100381610038161003816

119901d119904)1119901

le 119888119879120578(int

119905

0

(1 +1003816100381610038161003816120595(119904)

1003816100381610038161003816

119901)d119904)

1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817

119901

infin)1119901

le 119888119879120578+1119901

(1 +10038171003817100381710038171205951003817100381710038171003817infin)

(103)




E [10038161003816100381610038161003816119885119903119905(119909) minus 1198851199031015840 119905(1199091015840)10038161003816100381610038161003816

119901

]

le 119888 (1 + |119909|119901or10038161003816100381610038161003816119909101584010038161003816100381610038161003816

119901

)

times (10038161003816100381610038161003816119903 minus 119903

101584010038161003816100381610038161003816

119901120578

+10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816

119901

)

(104)


E [1003816100381610038161003816119885119903119905(119909)1003816100381610038161003816

119901] le 119888 (1 + |119909|

119901) 119890

119888119879120578119901+1

(105)





119903119904(120596 119885

119904119905(119909)) and

(120596 119903) 997891rarr 119885minus1

119903119905(120596 119909) belong to L

1199011



nabla119906119885119903119905= minus

119879(120590 ∘ 119885

1199051[119903119905]) (119906)

minus int

119905

119903

119879(120590

1015840(119885

119905) nabla

1199061198851199051[119903119905]) (119904) d (119904)

(106)


120585119872= inf 120591 1003816100381610038161003816nabla119906119885120591119905

1003816100381610038161003816

119901ge 119872 119885

119872

120591119905= 119885

120591or120585119872119905 (107)



E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

]

le 119888E [10038161003816100381610038161003816119879 (120590 ∘ 119885119872

1199051[119903119905]) (119906)

10038161003816100381610038161003816

119901

]

+ 119888E [int119905

119903

10038161003816100381610038161003816119879(120590

1015840(119885

119872

119905) nabla

119906119885119872

1199051[119903119905]) (119904)

10038161003816100381610038161003816

119901

d119904]

le 119888 (1 + E [int119905

119903

119906119901120578int

119906

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591 d119906]

+E [int119905

119903

119904119901120578int

119904

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591 d119904])

le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901(119905119901120578+1

minus 120591119901120578+1

) d120591]

+E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

(119905119901120578+1

minus 120591119901120578+1

) d120591])

le 119888119905119901120578+1

(1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591] + E [int119905

119903

10038161003816100381610038161003816nabla119906119885119872

120591119905

10038161003816100381610038161003816

119901

d120591])

(108)


E [10038161003816100381610038161003816nabla119906119885119872

119903119905

10038161003816100381610038161003816

119901

] le 119888 (1 + E [int119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (109)


E [1003816100381610038161003816nabla1199061198851199031199051003816100381610038161003816

119901] le 119888 (1 + E [int

119905

119903

10038161003816100381610038161198851205911199051003816100381610038161003816

119901d120591]) (110)




(120596 119903) 997891rarr 119885119903119904(120596 119885

119904119905(119909)) belongs to L

1199011



1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199011



minus1

119903119905(120596 + 119905ℎ 119909))



1

119905(119885

minus1

119903119905(120596 + 119905ℎ 119909) minus 119885

minus1

119903119905(120596 119909))

119905rarr0

997888997888997888997888rarr int

119879

0

119863119885minus1

119903119905(119904) ℎ (119904) d119904

(111)



(ΩL2)

We first show that

E[10038161003816100381610038161003816100381610038161003816

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909))

10038161003816100381610038161003816100381610038161003816

minus119901

] is finite (112)

Since

120597119885119903119905

120597119909(120596 119909)

= Id + int119905

119903

119879(120590

1015840(119885

119905 (119909))120597119885

119905 (120596 119909)

120597119909) (119904) d (119904)

(113)

letΘ119907= sup



E [Θ2119902

119907]

le 119888 + 119888 E[int119905

119906

Θ2(119902minus1)

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

d119904]

+ 119888 E[(int119905

119906

Θ119902minus2

119904(int

119904

119906

1003816100381610038161003816120597119909119885120591119905 (119909)1003816100381610038161003816

119901| d120591)

2119901

)

2

d119904]

(114)

Hence

E [Θ2119902

119907] le 119888 (1 + int

119905

119907

E [Θ2119902

119904] d119904) (115)


minus1



nabla119885119903119905(120596 119885

minus1

119903119905(119909)) +

120597119885119903119905

120597119909(120596 119885

minus1

119903119905(119909)) nabla119885

minus1

119903119905(120596 119909) (116)


minus1

119903119905(119909))

100381710038171003817100381710038171199011le 1198881003817100381710038171003817119885119903119905(119909))

100381710038171003817100381721199011

10038171003817100381710038171003817(120597119909119885119903119905(119909))

minus1100381710038171003817100381710038172119901 (117)

hence 119885minus1119903119905

belongs to L1199011

Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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