SPDE

STOCHASTIC DIFFERENTIAL EQUATIONS WITH PARTIALDERIVATIVES.

Gikhman, IlyaPLLS49 East Fourth StreetCincinnati, OH 45202ph: (513) 763-8308e-mail: [email protected]

INTRODUCTIONStochastic partial differential equations are the part of the general theory of random fields

which is actively being developed and applied. There exist differentapproaches to setting andsolving these problems. We single out the principle ones. In monograph [39] the linear theorybased on the interpretation of parabolic operators as infinitesimal operators of diffusionprocesses is given. Generalizations of that approach were established in [33]. In thismonograph using the stochastic characteristic systems like the case of the first orderdeterministic partial differential equations it was shown that afirst and the second ordersstochastic partial differential systems can be solved. Using A number of application, where thenecessity of solving stochastic parabolic problems appears, havebeen shown. In [32,36] thegeneralization of direct methods of mathematical physics whichallows to prove solvability ofnon-linear stochastic systems is shown. For these methods itis necessary to obtain aprioriestimates assuming coefficients to be coercive, and also to prove the possibility of limittransition in finite-dimensional approximations of input equationto be monotone. The mainrole in these methods is assigned to a drift coefficient. When itis equal to zero, a diffusioncoefficient can be only a bounded operator, that satisfies a Lipschitz condition. In [41]martingale statement is given and solvability of a number of evolution problems is proved. It iswell known that the existence of a finite second moment is essential for the existence ofsolutions of ordinary stochastic differential equations. Under certain assumptions oncoefficients growth, as it is shown in [21], a solution of an ordinary stochastic differentialequation does not possess a finite second moment. In [22, 28] a solutionof parabolic equationswith coefficients of ”white noise” type is constructed. One more approach to the solution ofstochastic parabolic problems consists in the substitution of integral equation of Ito-Volterratype in corresponding functional spaces for initial problem [3, 6, 34]. This method has beeninitial in stating and solving stochastic equations with unbounded operators.

We have mentioned only the works in which the principal approaches are described. Atpresent the list of papers and books where solutions of stochasticevolution problems arestudied is quite long and covers also physical, chemical and biological literature.

In this paper a direct probabilistic method of a solution of the Cauchyproblem forsemi-linear parabolic equation is suggested, and its physical interpretation is considered. Someof the results presenting here are given in [15]. We note that the results obtained belowcorrespond to those obtained by probabilistic methods used for a study of deterministicquasilinear parabolic equations [4, 11, 40]. It is also worthy to note thedifference between the

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solutions of semilinear and quasilinear systems, although the semilinear case of determinedsystems has not been taken separately. The latter allows to prove the existence and uniquenessof the ”classical” solution of the Cauchy problem when coefficients are sufficiently smooth.The former makes this possible only for a continuous solution [11, 40].Moreover, the solutionis local in character when there is no dissipation.

At the last part of the paper we introduce mathematically correct stochastic interpretationof the Schrodinger equation solution [16-18,23-26] and some model examples that use thisrepresentation. The same complex-valued representation for theShrödinger equation solutionfirst was given in [9]. Some applications of this approach to the particular problems werestudied in [1,8,31].

Now, we shall remind the basic aspects of the Lagrange and Euler formalisms used todescribe the dynamics of solid medial. Further, physical interpretation will be generalized forthe stochastic case. We can describe dynamic processes in physical media in two ways. Thefirst consists in the treatment of medium parameters, for each moment t ≥ 0, as functions ofsome fixed system of coordinates x1, x2, ..., xn . Such a method is called Euler method,and coordinates are Euler coordinates. The second is called theLagrange method. It consists inthe interpretation of solid medium as an aggregation of particles. Each particle differs from theother ones in its initial position. Both of these alternativeapproaches play an important role forstatistical description of motion in nonhomogeneous media, in the turbulence theory and otherapplications [22]. We shall now analyze these methods in the main.It is convenient to interpretparticle motion trajectory via the inverse of time.

Let, for the initial moment t ≥ 0 , a particle occupy the position ofy and supposeV s, x to be a velocity of this particle at the point x and moments ≥ 0. Then the Eulercoordinate x = x s ; t , y at the moments ∈ 0, t is calculated by the formula

x i s ; t, y = y i −t

s

∫ v i r, x r ; t, y d r , i = 1, 2, ...n #

Formula ( 0. 1 ) transforms the Lagrange coordinates into the Euler ones. Assume thatF t, x is a smooth function of the Euler coordinates and a medium parameter. It is easy torepresent it as a function of the Lagrange coordinates

Fl s, y = F s, y −t

s

∫ v i r, x r ; t, y d r #

By differentiating the relation ( 0. 2 ) with respect tos we find

d F l s, y

d s=

∂ F s, x

∂ s+

n

i = 1

∑ v i s, x ∂ F s, x

∂ x ix = x s, t, y

#


The expression in square brackets on the right hand side of ( 0. 3 ) is total derivative withrespect to time along particle path. It is called a substantive derivative. In the case, when thesolution of equation ( 0. 1 ) is unique and the functionV t, x is sufficiently smooth,formula ( 0.1 ) effects diffiomorfism of the Lagrange coordinates into Euler ones. In order toshow the equivalence of the Lagrange and Euler formalisms we construct the inverse mappingof the Euler coordinates into Lagrange ones. It means, by definitions to find a sufficientlysmooth functionG s, x for which the equation

G s, x s ; t , x = y #

has a unique solution. Taking into account of the formula for a substantive derivative, one

can write equation ( 0. 4 ) in the form

G 0, y +t

s

∫ ∂∂ s

+ v s, x , ∇ G s, x |x = x s ; t , y

d s = y

#

where ∇ = ∂∂ x 1

, ∂∂ x 2

, ..., ∂∂ x n

. From this it follows that the

function G t, x , which defines the transformation is a solution of the Cauchy problem

∂ G s, x

∂ s+

n

i = 1

∑ v i s, x ∂ G s, x

∂ x i

= 0 , G 0, y = y #

We next consider the case when particle paths are described by solutions of the ordinarystochastic differential equations. To do this an accurate theoryanalogous to the formalismdescribed above can be establish. It will turn out, that the definition of the functionG ∗ , ∗ along the trajectory of the solution of stochastic differential equation ( 0. 4 )can be represented by solution of ( 0. 4 ) in the Euler coordinates withrespect tomacroparameterG t , x which characterizes solid medium. We note that the equationsused here, with inverse time for the Lagrange trajectories presuppose to consider the Eulerdynamics via a straight direction of time.

Semilinear SPDE .

Before representing our results, let us introduce another approach to the Cauchy problemstudy of the SPDE of the parabolic type. After the presentation of the Backward StochasticDifferential Equations ( BSDE ) by Peng and Pardoux [37] in 1990, this field has receivedincreasing interest and activity [10,37,38]. Following [38] let


d Xst,x = b s , X

st,x ds + σ s , X

st,x d w s

wheres ∈ t , ∞ be a classical Ito equation solution such thatXtt,x = x. The

associated Peng-Pardoux BSDE is

Yst,x = h X

Tt,x +

T

s

∫ f r , Xrt,x, Y

rt,x, Z

rt,x dr +

+T

s

∫ g r , Xrt,x, Y

rt,x, Z

rt,x d B r −

T

s

∫ Zrt,x d w r

wheres ∈ t , T , and w t , B t , t ≥ 0 are two independent Wienerprocesses. They proved that under certain assumptions on coefficients of BSDE ,X

st,x has a

version whose trajectories are continuous int, s and twice continuously differentiable inx.They showed that the random fieldZ

st,x has a modification such that

Zst,x = ∇ Y

st,x ∇ X

st,x − 1 σ X

st,x

Thus, substituting this value into BSDE and letting

Yrt,x = u r, X

rt,x and then Y

tt,x = u t, x .

we can see that BSDE is the type of the stochastic evolution equation that is studied in thissection. An unessential difference between BSDE and the stochastic evolution systems studiedearlier is direction of time. Peng-Pardoux chose forward time for the characteristicsX ∗ andthus implied inverse time for BSDE. In papers [15,19,19,22] were used backward time for thecharacteristics and forward time for the evolution which gives us the possibility of studyCauchy problem for SPDE in the common direction of time. It seems also important toemphasize that the application BSDE to the study of the deterministic PDE [10] in theirexplicit evolutionary form will lead us to the probabilistic methodsof the study Cauchyproblem for the nonlinear parabolic deterministic systems [4,7,11,27,40].

We will introduce the notations. LetEm

be -dimensional Euclid space andE + = 0, + ∞ . Assume that the nonrandom Borel functionsb t, x , C t, x ,

defined for t , x ∈ E + × Em

and taking values inEm

and Ed

× Em

satisfy thefollowing conditions

| b t, x | + | C t, x | ≤ K 1 + | x | #

| b t, x − b t, y | + | C t, x − C t, y | ≤ | x − y |

where K > 0, | C |2

= Tr CC∗. Consider the Ito SDE with inverse time


ξ s ; t, x = x +t

s

∫ b r, ξ r ; t, x d r −t

s

∫ C r, ξ r ; t, x d←w r

#

Here, the stochastic integral is the inverse stochastic Ito integral, andw t is d-dimensionalWiener process on a some complete probability space Ω , Ϝ , P . By definition thesolution of equation ( 1. 2 ) is a random functionξ s ; t, x measurable in its variabless, t, x, ω respectively, and for fixedt, x with respect to theσ - fields

Ϝs

= σ w t′ − w t

′′ ; t

′, t

′′ ∈ s, t

and satisfying the equality ( 1. 2 ) with probability 1 for alls, t, x at once. Taking intoaccount the fact that SDE theory with inverse time is identicalto the one with the standardtime, we are going briefly to state the properties for backward Itointegrals. Let us introducetwo two-parametersσ- fields

→Ϝ s

t,

←Ϝ t

Tare mutually independent of each other, right and

left hand sides continuous respectively and

→Ϝ s

t 1 ⊂→Ϝ s

t 2 ⊂ Ϝ ;←Ϝ t 2

T ⊂←Ϝ t 1

T ⊂ Ϝ ; s ≤ t 1 ≤ t 2 ≤ T

The upper arrow points on time direction forϜ. Put

Ϝ t =→Ϝ 0

t ∪←Ϝ 0

tand Ϝ

t=

→Ϝ t

T ∪←Ϝ t

T.

Let w t , Ϝ t

T, t ∈ 0 , T be a Wiener process with the values inE d, g t

be a randomϜt − measurable d× n matrix-function . Define backward stochastic Ito

integral

←

I =T

0

∫ g t d←

w t

We will show that integral←

I possesses standard Ito integral properties. First suppose

thatg t is a step-function

g t =n − 1

i = 0

∑ g t i + 1 χ t i , t i + 1 t

where 0 = t 0 < t 1 < ... < t N = T is a partition of the interval 0, T . Setting by definition


←

I =n − 1

i = 0

∑ g t i + 1 ←w t i −

←w t i + 1

1) Ift ≤ T

sup E | g t | < ∞ , then E←I = 0. In fact

E←

I =n − 1

i = 0

∑ E E g t i + 1 ←w t i −

←w t i + 1 / Ϝ

t i + 1 =

=n − 1

i = 0

∑ E g t i + 1 E ←w t i −

←w t i + 1 / Ϝ

t i + 1

So as an increment←w t i −

←w t i + 1 is

←Ϝ t i

t i + 1 − measurable and independent

on Ϝt i + 1 then conditional expectation coincide with expectation and henceequal to 0.

2) If ∫ 0

TE | g t |

2d t < ∞ , then E |

←I |

2 ≤T

0

∫ E | g t |2

d t.

Really, for i < j

E g t i + 1 ←w t i −

←w t i + 1

∗g t j + 1 ×

× ←w t j −

←w t j + 1 = E E

←w t i −

←w t i + 1

∗/ Ϝ

t i + 1

g∗ t i + 1 g t j + 1

←w t j −

←w t j + 1 = 0

Summing over alli , j note that

E |←I |

2=

n − 1

i = 0

∑ E tr g∗ t i + 1 g t i + 1 t i + 1 − t i =

= ∫ 0

TE | g t |

2d t


3) For any positive numbersN and C

P |T

0

∫ g t d←w t | > C ≤ N

C2 + P

T

0

∫ | g t |2

d t ≥ N

Set g N t = g t χ t

0

∫ | g s |2

d s ≤ N . Then

P t ≤ T

sup | g N t − g t | > 0 = P T

0

∫ | g t |2

d t ≥ N .

Hence,

P |T

0

∫ g t d←w t | > C ≤ P |

T

0

∫ g N t d←w t | > C +

+ P |T

0

∫ g t − g N t d←w t | > 0 ≤ C

2E |

T

0

∫ g N t d←w t |

2+

+ P T

0

∫ | g t |2

d t ≥ N ≤ NC

2 + P T

0

∫ | g t |2

d t ≥ N

Following standard ideas it is easy to generalize these properties for anyϜt −

measurable random functions such that∫ 0

T| g t |

2d t < ∞ with probability 1.

Taking into account that SDE theory with the inverse time is identical to that one with thestandard time , we shall state the result.

Theorem 1.1. Assume that coefficients of equation (1. 2) satisfy the conditions (1.1).Then there exists a unique solution of equation (1.2), such that for anyq ≥ 2 the followingestimates are valid

s , t

sup E | ξ s ; t, x |q ≤ L 1 L 2 + | x |

q ,


s , t

sup E | ξ s ; t, x − ξ s ; t, y |q ≤ L 3 | x − y |

q, #

s , t

sup E | ξ s ; t, x − x |q ≤ L 4 L 5 + | x |

q ,

where L j , j = 1, 2 ... are positive constants dependent onT, k, q . Along thepaths of the solutions of the system (1.2) we consider equation

u t, x = ϕ ξ s ; t, x +t

s

∫ f r, ξ r ; t, x , u r, ξ r ; t, x d r −

−t

s

∫ g r, ξ r ; t , x d←w r #

The stochastic integral on the right-hand side of equation (1.4) is interpreted as the inversestochastic Ito integral. Here we just turn to the case, when the right-hand side of the equality(1.4) depends onξ ∗. The equation (1.4) is a generalization of the (0.2), where initial datafor Euler coordinates of the particleξ ∗ ; t, x are the Lagrange variables of thefunctions u t, x . Note that the processξ s ; t, x in equality (1.4) plays the samerole as characteristics in the deterministic hyperbolic systems of the first order. As a solution ofequation (1.4), we understand the separable random functionu t, x measurable in itsvariables and for anyx adapted to a filtration

Ϝ t = σ w t′ − w t

′′ ; t

′, t

′′ ∈ 0, t

and satisfying the equality (1.4) with probability 1 for allt, x at once.Theorem 1.2. Assume that non-random continuous int ∈ 0, T coefficients of the

system ( 1. 2 ), ( 1, 4 ) satisfy the conditions (1.1) and the functions

ϕ x , f t, x, u , g t, x defined for t, x, u ∈ 0, T × Em× E

nand

taking values inEn

, En

, Ed

× En

respectively, satisfy the conditions

| ϕ x | + | f t, x, u | + | g t, x | ≤ k 1 + | u |


u ≠ v

t

x ≠ y

sup | ϕ x − ϕ y |

| x − y |γ +

| f t , x , u − f t , y , v |

| x − y |γ

+ | u − v |+

#

+| g t , x − g t , y |

| x − y |γ < ∞

where γ ∈ 0 , 1 . Then equation (1.4) has a unique solution for which

|| u || 2,γ =t, x

sup E | u t, x |2

12 +

x ≠ y

t ≤ T

sup E| u t, x − u t, y |

2

| x − y |γ

12

+

+t ≠ s , x

sup E| u t , x − u s , x |

2

| t − s |γ2 1 + | x |

12

< ∞ .

Proof. We denote byM a set of bounded infinitely differentiable int , x withprobability 1, random functionsu t, x, ω measurable with respect toϜ t , t ≥ 0 foreach x. Then there exists a separable modification for each function from M that may takeinfinite values and an arbitrary set in 0, T can serves as its separability set. Identifying theclass of stochastically equivalent functions with the modification, we fix one separability set forall functions from M . Identifying the class of stochastically equivalent functionswith theseparable modification, we fix one and the same separability set for all functions fromM .Denote B 2 , γ the Banach space obtained as completion of the setM in norm ||u || 2 , γ . It is

obvious thatϜ t -measurable random functions continuous int , x serve as elements ofspaceB 2 , γ . Let ( Z u t, x, ω be the right-hand side of equality (1.4) and

v t, x, ω be an arbitrary function fromB 2 , γ . It is easy to verify that

|| Z v || 2 , γ < ∞ . It implies that the operatorZ is defined as an operator acting from

spaceB 2 , γ into itself. Then the series

∞

i = 1

∑t , x

sup E | Zi + 1

v t, x − Zi

v t, x |2

converges and, consequently, there exists a limit of the functions Ziv t, x, ω as

i → ∞ for fixed t , x with probability 1. Denote this limit byu t, x, ω and it iseasy to check that


Z u t , x , ω = u t , x , ω

and ||u || 2 , γ < ∞ . Thus the functionu t, x, ω belongs toB 2 , γ there exists

modification for which equation Z u t, x, ω = u t, x, ω holds for all t, x atonce with probability 1. The uniqueness of the solution of equation (1.4) follows from the factthat a certain power of the operatorZ is contraction operator in Banach spaceB 2 which

completes setM in the norm

|| u || 2 =t , x

sup E | u t, x |2

12

In proving Theorem 2 the same separability set for all elements of the B 2 , γ is used.

Further it is convenient to fix this set.Theorem 1.3.Assume that the functionsb i , b , C i , C satisfy the conditions of

theorem 1.1, and the functionsϕ i , ϕ, f i , f , g i , g satisfy the conditions of theorem 1.2with γ = 1 and

i → ∞

limT

0

∫x

sup | b i t, x − b t, x | + | C i t, x − C t, x | +

+ | g i t, x − g t, x | +u

sup | f i t, x, u − f t, x, u | +

+ | ϕ i x − ϕ x | d t = 0

Then, for any q ≥ 2

i → ∞

limt , x

sup E | u i t, x − u t, x |q= 0 .

Here u i t, x is a solution of the systems (1. 2) and (1.4) with coefficientsb i , c i , ϕ i , f i , g i .

The proofs of the theorems 1.3,1.4 can be obtained by standard methods usingthe results[12, 19] .

Denote by Cq + γ

a set of non-random functions, whoseq -order derivatives satisfyHolder condition withγ ∈ 0, 1 .

Theorem 1.4.Assume that coefficients of the system ( 1. 2 ) and ( 1. 4 ) belong to the

space C1 + γ ∩ C

2 + γin variablesx andu. Then


x ≠ y, t

sup E |∇ 2 u t , x − ∇ 2

u t , y

| x − y |γ |

2< ∞.

Consider now the solution of the problem ( 1. 4 ) as a functional of theparameters. Denoteby u t , x ; s , ϕ , t ≥ s the solution of the problem (1.4) for which

u s , x ; s , ϕ = ϕ x .Lemma 1.1.Suppose the conditions of theorem 2 hold andu t, x ; s, ϕ, ω is the

solution of equation ( 1. 4 ). Then for arbitraryr ∈ s , t for alls ∈ 0 , t , t ∈ 0 , ∞ , x ∈ E

nwith probability 1

u t , x ; s , ϕ , ω = u t , x ; r , u r , ∗ ; s , ϕ , ω , ω

Proof. The random functionu t , x ; s , ∗ , ω is separable and continuous in its

variables t , x , s and the random functionsu t , x ; r , ψ , ω andu r , x ; s , ϕ , ω are independent for anyr ∈ s , t and any for nonrandomfunctions ϕ , ψ , since the functionu t , x ; r , ψ , ω , for nonrandomψ, dependsonly on the increments of the Wiener process on the interval r , t , and functionu r , y ; s , ϕ , ω depends only the increments of this process over the interval s ,r . It immediately follows from equality (1.4) that

u t, x ; s, ϕ, ω = ϕ ξ s ; r , y +

+s

r

∫ f l, ξ l ; r, y , u l, ξ l ; r, y ; s, ϕ d l

−r

s

∫ g l, ξ l ; r, y d←w l y = ξ r ; t, x +

+t

r

∫ f l, ξ l ; t, x , u l, ξ l ; t, x ; s, ϕ, ω dl −

−t

r

∫ g l , ξ l ; t, x d←w l = u r, ξ r ; t, x ; s, ϕ, ω +

+t

r

∫ f l, ξ l ; t, x u l, ξ l ; t, x ; s, ϕ d l −


−t

r

∫ g l , ξ l ; t , x d←w l

Theorem 3 implies that the functionu t, x ; s , ϕ , ω is continuous inϕ . It iseasy to observe that the function satisfies the equation

u t , x ; r , u r , ∗ ; s , ϕ = u r, ξ r ; t , x ; s , ϕ +

+t

r

∫ f l, ξ l ; t, x , u l, ξ l ; t, x ; r, ϕ dl −

−t

r

∫ g l, ξ l ; t, x d←w l

By subtracting this equality from the preceding one, we find

xsup |Δ u t , x | < K

t

s

∫x

sup |Δ u r , x | d r

where Δ u t , x = u t, x ; s, ϕ − u t, x ; r, u r, ∗ ; s, ϕ and K is aLipschitz constant of a functionf with respect to u. Using Gronwall’s lemma we complete theproof.

Theorem 1.5.Assume that the coefficients of equation ( 1 .2 ) are continuous int andfor each t , second derivatives of the coefficients of the system ( 1. 2 ), (1. 4 ) satisfyHolder’s condition withγ ∈ 0 , 1 in variablex and u are uniformly bounded. Thenthe solution of the problem (1. 4) is also a solution of the followingCauchy problem

u t , x = ϕ x +t

0

∫ m

i = 1

∑ ∂ u s, x

∂ x i

b i s, x +

+ 12

m

i, j = 1

∑d

k = 1

∑ ∂ 2u s, x

∂ x i ∂ x j

c i k s, x c j k s, x + f s, x, u d s +

#


+m

i = 1

∑d

k = 1

∑t

0

∫ c i k s , x ∂ u s , x

∂ x i

d w k s +t

0

∫ g s, x d w s

for all t, x with probability 1. Here, the stochastic integrals areinterpreted in a ”classical”sense, i. e. as Ito’s integrals with the standard time.

Before proving the theorem, we illustrate its applications. First let us remark that equation( 1. 6 ) is the equation in Lagrange coordinates. Using this equation, it is easy to construct thefunctionG t , x , which transform the Euler coordinates into Lagrange coordinates.Indeed, letG t , x = G t , x, ω be a smooth random function. Let

0 = t 0 < t 1 < t 2 < .... < t N = t be a partition of the interval 0 , t andλ =

i

max t i + 1 − t i . Then

G t , y − G 0 , ξ 0 ; t , y =

=

i = 0

N − 1

∑ G t i + 1 , ξ t i + 1 ; t , y − G t i , ξ t i ; t , y =

=

i = 0

N − 1

∑ ∂ G t i , ξ t i ; t , y

∂ tΔ t i + ξ t i + 1 ; t , y − ξ t i ; t , y ⋅

∇ G t i , ξ t i ; t , y +1

2tr ξ t i + 1 ; t , y − ξ t i ; t , y ∗ ⋅

ΔG t i , ξ t i ; t , y ξ t i + 1 ; t , y − ξ t i ; t , y + α λ ω

where α λ ω = 0. Taking the limits of all summations asλ → 0 in the formula weget

G t, y − G 0, ξ 0 ; t , y =

t

0

∫ ∂ G s, ξ s

∂ t−

− b s, ξ s ; t, y ∇ G s, ξ s ; t, y +

+ 12

Tr C∗ s, ξ s ; t, y Δ G s, ξ s ; t, y C s, ξ s ; t, y d s


−t

0

∫ C∗ s, ξ s ; t, y ∇ G s, ξ s ; t , y d

←w s

In order to find a function that transforms the Euler coordinatesinto Lagrange coordinates,we put G 0, ξ 0 ; t , y = y. Then

y = G t, y −t

0

∫ ∂ G s, ξ s; t, y

∂ s−

− b s, ξ s ; t, y ∇ G s, ξ s ; t, y +

+ 12

Tr C∗ s, ξ s ; t, y Δ G s, ξ s ; t, y C s, ξ s ; t, y d s +

+t

0

∫ C∗ s, ξ s ; t, y ∇ G s, ξ s ; t , y d

←w s

Thus, let G s , x, ω be a formal solution of the inverse Cauchy problem

∂ G s, y

∂ s− b s, y ∇ G s, y + #

+ 12

Tr C∗ s, y ΔG s, y C s, y + C

∗ s, y ∇G s, y

d←w s

d s= 0 ,

s ∈ 0, t with boundary condition on the end of the span

G t , y = y

Then G 0, ξ 0 ; t , y = y and thereforeG t , y, ω transforms Eulercoordinates into Lagrange’s. The question concerning the construction of the first integrals ofthe solution of equation ( 1. 2 ) can serve as another application of Theorem 1.5. Recall that thefunctionV t , x is a first integral of the solution of the equation (1.2), if for alls ∈ 0 , t , V s , s , ξ s ; t , y = const. Analogously to what has been

said above, it is easy to observe that the first integrals satisfy equation (1.7) with initialcondition. V t , x = const. The first integrals were considered in [7, 39].

Proof of theorem 1.5. Let s = t 0 < t 1 < t 2 < .... < t N = t be a partitionof the interval s , t . Then

u s, ξ s ; t, y − u s, y =


=

i = 1

N

∑ u s, ξ t i − 1 ; t, y − u s, ξ t i; t, y

In order to simplify our notations, we setξ s = ξ s ; t , y . Since the functionu t, x is twice continuously differentiable with probability 1, and using Taylor’s formula

we have

N

i = 1

∑ u s, ξ t i − 1 − u s, ξ t i =

=

i

∑ u 1

s, ξ t i ξ t i − 1 − ξ t i +

+ 12

Tr ξ t i − 1 − ξ t i ∗

u 2

s, ξ t i ξ t i − 1 − ξ t i +

+ 12

1

0

∫ 1 − l Tr ξ t i − 1 − ξ t i ∗ u

2 s, ξ l t i −

− u 2

s , ξ t i ξ t i − 1 − ξ t i d l = I 1 + I 2 + I 3

Here, u 1

t , x = ∇ u t , x , u 2

t , x = ∇ 2 u t , x, andξ l t i = l ξ t i − 1 + 1 − l ξ t i . We shall now show thatI 3 tends to

0 in probability for λ = max t i − t i − 1 → 0 . By Chebyshev’s inequality for

arbitrary ε > 0

P | I 3 | > ε ≤ 2 ε − 1

i = 1

N

∑ E

1

0

∫ ξ t i − 1 − ξ t i 2×

× u 2

s, ξ l t i − u 2

s, ξ t i d l ≤ 2 ε − 1| u 2

| 1,γ ×


×N

i = 1

∑ E1

0

∫ ξ t i − 1 − ξ t i 2 + γ ≤ R ε

− 1 t − s λ

γ2 ,

where R is some constant, dependent onT and on coefficients of the equation. We nowestimate the termsI 1 and I 2 . We have

I = ∑ i u 1

s , ξ t i ξ t i − 1 − ξ t i =

= ∑ i u 1

s , ξ t i b t i , ξ t i t i − t i − 1 +

+ C t i , ξ t i w t i − 1 − w t i + o 1 λ

where

o 1 λ =N

i = 1

∑ u 1

s , ξ t i

t i

t i − 1

∫ b r, ξ r −

− b t i , ξ t i d r +

t i

t i − 1

∫ C r, ξ r − C t i , ξ t i d←w r |

Then

E | o 1 λ |2 ≤ 2 λ || u

1 ||

2

| s 1 − s 2 | ≤ λ

sup E | b s 1 , ξ s 1 −

− b s 2 , ξ s 2 |2 t − s + 2 E | ∑ i u

1 s , ξ t i ×

×

t i

t i − 1

∫ C r, ξ r − C t i , ξ t i d←w r |

2.

Estimating the second term we seti < j . Since random variables


t j

t j − 1

∫ C r , ξ r − C t i , ξ t i d←w r and ξ t j

are Ϝt i − measurable, then

E | u 1

s, ξ t i

t i

t i − 1

∫ C r, ξ r − C t i , ξ t i d←w r ×

× u 1

s, ξ t j

t j

t j − 1

∫ C r, ξ r − C t j , ξ t j d←w r | =

= E

t i

t i − 1

∫ C r, ξ r − C t i , ξ t i d←w r

∗×

× E u 1

s, z j ∗

u 1

s, z i ×

×

t j

t j − 1

∫ C r, ξ r − C t i , ξ t i d←w r / Ϝ

t i z j = ξ t j

z i = ξ t i

The expression under the sign for the conditional mathematical expectation, doesn’t

depend onσ − algebraϜt i . Hence the conditional mathematical expectation considers

with the unconditional one. Then

E u 1 s, z j ∗u 1 s, z i

t i

t i − 1

∫ C r, ξ r − C ti , ξ t i d←

w r =

= E E u 1

s, z j ∗

u 1

s, z i

t i

t i − 1

∫ C r, ξ r −


− C t i , ξ t i d←w r / Ϝ s = E u

1 s, z j

∗u

1 s, z i ×

× E

t i − 1

t i

∫ C r, ξ r − C t i , ξ t i d←w r = 0

Thus the second term of the estimate of the variable doesn’t exceed

E

i

∑ | u 1

s, ξ t i |2|

t j

t j − 1

∫ C r, ξ r − C t i , ξ t i d←w r |

2 ≤

≤ || u 1

||2

sup E | C s 1, ξ s 1 − C t 2 , ξ t 2 |2

t − s

wheresup is taken over |s 1 − s 2 | < λ . Hence

E | o 1 λ |2 ≤ 2 λ || u

1 ||

2 t − s

| s 1 − s 2 | < λ

sup E λ | b s 1, ξ s 1 −

− b s 2 , ξ s 2 |2+ | C s 1, ξ s 1 − C t 2 , ξ t 2 |

2 .

By the conditions of the theoremλ → 0

lim E | o 1 λ |2

= 0 . Further

I 2 = 12

i

∑ Tr C∗ s, ξ t i u

2 s, ξ t i C s, ξ t i t i − t i − 1 +

+ o 2 λ

where

| o 2 λ | ≤ 12

i

∑ | u 2

s, ξ t i |2

t i

t i − 1

∫ | C r, ξ r C∗ r, ξ r −


− C t i , ξ t i C∗ t i , ξ t i | d r + |

t i

t i − 1

∫ b r , ξ r d r |2

Analogously to the estimateI 1 it is easy to observe thatλ → 0

lim E | o 2 λ |2= 0.

Thus

I 1 + I 2 =

=N

i = 1

∑ u 1

s, x b s, x ti − ti − 1 + C s, x w t i − 1 − w t i +

+ 12

N

i = 1

∑ Tr C∗ s, x u

2 s , x C s, x t i − t i − 1 +

+ o 1 λ + o 2 λ + o 3 t − s

where

o 3 t − s =N

i = 1

∑ u 1

s, ξ t i b t i , ξ t i −

− u 1

s, x b s, x t i − t i − 1 + u 1

s, ξ t i C t i , ξ t i −

− u 1

s, x C s, x w t i − 1 − w t i +

+ 12

N

i = 1

∑ Tr C∗ s, ξ t i u

2 s, ξ t i C s, ξ t i −


− Tr C∗ s , x u

2 s , x C s , x t i − t i − 1 .

We now estimate the sum in which the increments of the Wiener processw t arecontained. Then

P |

i

∑ u 1

s, ξ t i C t i , ξ t i − u 1

s, x C s, x ×

× w t i − 1 − w t i | > 2 ε ≤ P |

i

∑ u 1

s , ξ t i −

− u 1

s , x C t i , ξ t i w t i − 1 − w t i | > ε +

+ P |i

∑ u 1 s, x C t i, ξ t i − C s, x ×

× w t i − 1 − w t i | ≥ ε

The second summand on the right can be estimated by Chebyshev’s inequality. Since therandom functionsξ t i = ξ t i ; t , x , w t i − 1 − w t i

areϜs − measurable for anyi and u s , x is independent of theσ −algebra, the

second term doesn’t exceed

ε− 2

xsup E | u

1 s , x |

2

r ∈ s , t

sup |E | C r, ξ r C∗ r, ξ r −

− C t i , ξ t i C∗ t i , ξ t i |

2 t − s

Then

P | ∑ i u 1

s, ξ t i − u 1

s , x C t i , ξ t i ×

× w t i − 1 − w t i | > ε ≤ ε− 2

|| u 2

||2

K2×

×r ∈ s , t

sup |E | C r , ξ r − x |2 t − s ≤ R 1 t − s

2


where the positive constantR 1 is easily defined by formula (1.3). Estimate the remainedterms of theo 3 t − s . We have

P

i

∑ | u 1

s, ξ t i b t i , ξ t i − u 1

s, x b s, x +

+ 12 ∑ i Tr C

∗ s, ξ t i u

2 s, ξ t i C s , ξ t i −

− Tr C∗ s , x u

2 s , x C s , x | t i − t i − 1 > ε ≤

≤ ε− 1∑ E | u

1 s, ξ t i b t i , ξ t i − u

1 s, x b s, x |+

+ 12

Tr C∗ t i , ξ t i u

2 s, ξ t i C t i , ξ t i −

− Tr C∗ s, x u

2 s, x C s, x | t i − t i − 1 ≤

≤ ε− 1

R 2 t − s 12 +

r ∈ s , t

sup E | b r , ξ r − b s , x | +

+r ∈ s , t

sup E | C∗ r, ξ r C r, ξ r − C

∗ s, x C s, x | t − s

where R 2 ≥ 0 is some constant. Following the assumption of the theorem, weobserve

that

λ → 0

lim I 1 + I 2 + I 3 =

= u 1

s, x b s, x t − s + C s, x w s − w t

+ 12 ∑ i Tr C

∗ s, x u

2 s, x C s, x t − s + o t − s

where


P.t → s

lim t − s − 1

o t − s = 0 .

Let 0 = s 0 < s 1 < ... s N = t . Applying the estimates obtained above to eachinterval s i , s i + 1 we find

N − 1

i = 0

∑ u s i + 1, x − u s i, x =N − 1

i = 0

∑ u s i, ξ s i ; s i + 1, x − u s i, x +

+

s i + 1

s i

∫ f s, ξ s ; s i + 1 , x , u s, ξ s ; s i + 1, x d s −

−

s i + 1

s i

∫ g s , ξ s ; s i + 1 , x d←w s =

N − 1

i = 0

∑ u 1

s i , x b s i , x +

+ 12

Tr C∗ s i , x u

2 s i , x C s i , x s i + 1 − s i +

+N − 1

i = 0

∑ u 1

s i , x C s i , x w s i + 1 − w s i +

+ f s i , x, u s i , x s i + 1 − s i +

+ g s i , x w s i + 1 − w s i + o λ

Here λ = max s i + 1 − s i and P.λ → 0

lim | o λ | = 0 . Passing to the limit

in this equality as λ → 0 , we obtain the functionu t , x to be a solution of equation(1.6) and Theorem1.5 is proved.

Next we consider a generalization of theorem 1.5. Letw i t be independent Wiener

processes taking values inEd i , i = 1 , 2 , ... l . Denote by ϜΔ

i theσ −algebra


generated by the increments of thew i t , t ∈ Δ and Ϝ Δ =I

i = 1

∪ ϜΔ i . Consider

the system

ξ s ; t, x = x +t

s

∫ b r, ξ r ; t, x d r −i = 1

I

∑t

s

∫ C i r, ξ r ; t, x d←wi r

u t, x = ϕ ξ s ; t, x + #

+t

s

∫ f r, ξ r ; t, x , v r, ξ r ; t, x d r −t

s

∫ g r, ξ r ; t, x d←

w1 r

where v t , x = E u t , x | ϜΔ 1 and C i t , x are nonrandom

matrices with elementsc k j

i t , x ; k = 1, 2, ...d i , j = 1, 2 , .. m . It is easy

to show, that the results proved in theorem 1.1-4 hold for the system (1.8 ).Theorem 1.6. Assume that the coefficients of the system (1.8 ) satisfy the conditions of

theorem 1.5. Then the functionv t , x , ω is a ”classical” solution of the Cauchyproblem

v t , x = ϕ x +t

0

∫ b s , x , ∇ v s , x +

+ 12

C s, x C∗ s, x ∇ , ∇ v s, x + f s, x, v d s + #

+t

0

∫ C 1 s, x d w 1 s , ∇ v s, x +t

0

∫ g s, x d w 1 s

where ,

C s , x C∗ s , x =

I

i = 1

∑ C i s , x C i

∗ s , x

and stochastic integrals on the right in (1.9) are interpreted in the classical Ito sense. Thethe proof of the theorem 1.6 follows that of theorem 1.5 in the main.Therefore we underlineonly the moments, which differ. We have


u t , x − ϕ x =N − 1

j = 0

∑ u s j + 1 , x − u s j , x =

=N − 1

j = 0

∑I

i = 1

∑ C i s j , x Δ w i s j ,∇ u s j , x +

N − 1

j = 0

∑ b s j , x , ∇ u s j , x +

+ 12

C∗ s , x C s , x ∇ , ∇ u s , x s j + 1 − s j +

+N − 1

j = 0

∑ f s j , x, v s j , x s j + 1 − s j +

+N − 1

j = 0

∑ g s j , x Δ w 1 s j + o λ .

Here 0 = s 0 < s 1 < ... s N = t is a partition of the interval 0 , t ,

Δ w i s j = w i s j + 1 − w i s j , λ = max s j + 1 − s j ,

and random variable o λ is such, that limλ → 0 | o λ | = 0 . Calculating the

conditional expectation with respect toσ − algebraϜ 0 , t

1and replacing the operator∇

and the conditional expectation, we have

E C i s j , x Δ w i s j , ∇ u s j , x / Ϝ 0 , t

1 =


= C i s j , x ∇ E u s j , x Δ w i s j / Ϝ 0 , t

1

We now consider the conditional mathematical expectation on theright-hand side of thisequality. Set

Ϝ 1 = Ϝ s j , t

1, Ϝ 2 = Ϝ 0 , s J

1, Ϝ 3 =

I

i = 2

∪ Ϝ s j , s j + 1

i ∪ Ϝ 0 , s j

It is easy to show thatσ − algebrasϜ 1 and Ϝ 3 are conditionally independent withrespect toσ −algebraϜ 2 . Let Y k are arbitrary nonnegative random variables , measurablewith respect to σ −algebraϜ k , k = 1, 3 . Taking into account thatϜ 2 ⊂ Ϝ 3 andthat σ −algebraϜ 1 does not depend onσ −algebrasϜ 2 and Ϝ 3 , we have

E Y 1 Y 3 / Ϝ 2 = E Y 3 E Y 1 / Ϝ 3 / Ϝ 2 =

= E Y 3 / Ϝ 2 E Y 1 = E Y 3 / Ϝ 2 E Y 1 / Ϝ 2 #

The equality obtained coincides with the definition of conditional independence ofσ −algebrasϜ 1 and Ϝ 3. Hence, (see[32]), with probability 1

E Y 3 / Ϝ 1 ∪ Ϝ 2 = E Y 3 / Ϝ 2 #

Assuming Y 3 = u s j , x Δ w i s j , for i = 2 , 3 , ... ,l we have

E u s j , x Δ w i s j / Ϝ 0 , t

1 =

= E u s j , x Δ w i s j | Ϝ 0 , s j

1 ∪ Ϝ s j , t

1 =

= E u s j , x Δ w i s j / Ϝ 0 , s j

1 =

= E u s j , x E Δ w i s j / Ϝ 0 , s j / Ϝ 0 , s j

1 = 0

In this casei = 1 we assume that


Ϝ 1 = Ϝ s j + 1 , t

1, Ϝ 2 = Ϝ 0 , s J + 1

1, Ϝ 3 = Ϝ s j , s j + 1

1 ∪ Ϝ 0 , s j .

Analogously to the preceding the equality (1.10) is valid and taking account of (1.11) wehave

E u s j , x Δ w 1 s j / Ϝ 0 , t

1 =

= E u s j , x Δ w 1 s j / Ϝ 0 , s j + 1

1 ∪ Ϝ s j + 1 , t

1 =

= E u s j , x / Ϝ 0 , s j + 1

1 Δ w 1 s j = v s j , x Δ w 1 s j

Hence, assumings j + 1 = t we obtain

E u s j , x / Ϝ 0 , t

1 = E u s j , x / Ϝ 0 , s j

1 = v s j , x

Therefore

E

i

∑ ∑j

C i s j , x Δ w i s j , ∇ u s j , x / Ϝ 0 , t

1 =

= ∑ j C 1 s j , x Δ w 1 s j , ∇ v s j , x ,

E N − 1

j = 0

∑ b s j , x , ∇ + 12

C∗ s j , x C s j , x ∇ , ∇ ×

× u s j , x Δ s j / Ϝ 0 , t

1 =

N − 1

j = 0

∑ b s j , x , ∇ +

+ 12

C∗ s j , x C s j , x ∇ , ∇ v s j , x Δ s j

Passing to the limit asλ → 0 , we can see that theorem 1.6 is proved.Now we shall study the inverse problem. Using equation (1.6), we shall find the equation

of particle trajectories. Moreover the uniqueness of the solution of the Cauchy problem (1.6) is


not supposed, but it follows from the uniqueness of the solution of the correspondingtrajectory- problems.

Let B 2

2 + γdenote the Banach space ofϜ t − measurable separable random

functions V t , x , ω with the norm

|| V || 2 , 2 + γ = max || ∇ iV || 2 , γ .

Theorem 1.7. Assume that V t , x , ω is an ’classical ’ solution of equation

(1.6) belonging to the spaceB 2

2 + γwith coefficients satisfy the conditions of the theorem 1.5.

Then the functionV t , x , ω is the solution of equation (1.4).Proof.Let the functionV t , x , ω be and arbitrary ’ classical’ solution of equation

(1.6) and s , t be an arbitrary subinterval of the interval 0 , T and ξ s ; t , x be a unique solution of equation (1.2) with the Wiener processw t that used in equation

(1.6). Then for an arbitrary functionQ t , x , ω ∈ B 2

2 + γwe have

Q s, ξ s ; t, x − Q t, x =

=N

i = 1

∑ Q s, ξ t i − 1 ; t, x − Q s, ξ t i ; t, x .

For the sake of simplicity of notations we assume further thatξ s = ξ s ; t , x .By Taylor’s formula with remainder term in the integral form:

N

i = 1

∑ Q s , ξ t i − 1 − Q s , ξ t i =

=N

i = 1

∑ ξ t i − 1 − ξ t i ∇ Q s , ξ t i +

+ 12

Tr ξ t i − 1 − ξ t i ∗Δ Q s , ξ t i ξ t i − 1 − ξ t i +

+1

0

∫ 1 − θ 12

Tr ξ t i − 1 − ξ t i ∗ Δ Q s, ξ θ t i −


− Δ Q s, ξ t i ξ t i − 1 − ξ t i d θ = I 1 + I 2 + I 3 ,

where ξ θ t i = 1 − θ ξ t i + θ ξ t i + 1 . For the function

Q t , x ∈ B 2

2 + γ, analogously to theorem 5 we obtain

P | I 3 | > ε ≤ R ε− 1

t − s λγ2 ,

where R is a constant independent ofx , ε , λ . For the expressionsI 1 and I 2 using

the methods applied above, we find

I 1 = ∑ i b t i , ξ t i t i − t i − 1 +

+ C t i , ξ t i w t i − 1 − w t i ∇ Q s, ξ t i + o 1 λ ,

I 2 = 12

i

∑ Tr C∗ t i , ξ t i Δ Q s, t i ×

× C t i , ξ t i t i − t i − 1 + o 2 λ

where

λ → 0

lim E | o 1 λ | + | o 2 λ | = 0 .

Hence, we can note that

I 1 + I 2 + I 3 +

+t

r

∫ f l, ξ l ; t, x , Q l, ξ l ; t, x d l −t

r

∫ g l, ξ l ; t, x d←w l =

=

i

∑ b s , x Δ t i + C s , x Δ w t i ∇ Q s , x +


+ 12

i

∑ Tr C∗ s, x ΔQ s, x C s, x Δ t i +

+ f s, x,Q s, x Δ t i + g s , x Δ w t i + o t − s

HereP.t → s

lim | o t − s | = 0 . Consequently,

Q s, x + b s, x t − s + C s, x w t − w s ∇ Q s, x +

+ 12

Tr C∗ s, x ΔQ s, x C s, x + f s, x,Q s, x t − s

+ g s, x w t − w s = Q s, ξ s + #

t

+s

∫ f l, ξ l ; t, x , Q l, ξ l ; t, x d l −

−t

s

∫ g l, ξ l ; t, x d←w l + o t − s

In the equality (1.12) which is valid for any functionQ from the spaceB 2

2 + γ, we choose

a given ’classical’ solutionV t , x , ω of the Cauchy problem (1.6) with initial conditionV 0 , x , ω = ϕ x . Let 0 = t 0 < t 1 < t 2 < .... < t N = t be apartition of the interval 0 , t , λ = max Δ t i . Then applying the equality (1.12) to t 0 , t 1 , we find

V t 1, x = ϕ ξ t 0 ; t 1, x +

+

t 1

t 0

∫ f l, ξ l ; t 1 , x , V l, ξ l ; t 1 , x d l −

−t 1

t 0

∫ g l, ξ l ; t 1, x d←

w r + o λ + γ t 1 − t 0 , x ,

where


γ t 1 − t 0 , x =

t 1

t 0

∫ b s , x ∇ V s , x +

+ 12

Tr C∗ s, x Δ V s, x C s, x + f s, x, V s, x −

− b t0 , x ∇V t0 , x + 12

Tr C∗ t0 , x ΔV t0, x C t0 , x +

+ f t 0 , x, V t 0 , x d s +

t 1

t 0

∫ C s, x ∇ V s, x + g s, x −

− C t 0 , x ∇ V t 0 , x + g t 0 , x d w s

Suppose, that for anyk the following representation holds

V t k − 1 , x = ϕ ξ t 0 ; t k − 1 , x +

+

t k − 1

t 0

∫ f l, ξ l ; t k − 1 , x , V l, ξ l ; t k − 1 , x d l −

−

t k − 1

t 0

∫ g l, ξ l ; t k − 1 , x d←w l +

+ k − 1 o λ +k − 1

i = 1

∑ γ t i − t i − 1 , x

We show that the similar representation also holds for the moment t k , and then estimate

the sum of variablesγ. By (1.12) we have

V t k , x = V t k − 1 , y +


+

t k

t k − 1

∫ f l , ξ l ; t k , x , V l , ξ l ; t k , x d l −

−

t k

t k − 1

∫ g l, ξ l ; t k , x d←w l + o λ + γ t k − t k − 1 , x =

= ϕ ξ t 0 ; t k − 1 , y +

+

t k − 1

t 0

∫ f l, ξ l ; t k − 1 , y , V l, ξ l ; t k − 1 , y d l −

−t 1

t 0

∫ g l, ξ l ; t k − 1 , x d←w l +

+k − 1

i = 1

∑ γ t i − t i − 1 , y y = ξ t k − 1 ; t k , x

+ #

+

t k − 1

t 0

∫ f l , ξ l ; t k , x , V l , ξ l ; t k , x d l −

−

t k

t k − 1

∫ g l, ξ l ; t k , x d←w l + k o λ + γ t k − t k − 1 , x

Since,

k − 1

i = 1

∑ γ t i − t i − 1 , y |y = ξ t k − 1 ; t k , x

+ γ t k − t k − 1 , x = 0

then, it is sufficient to show, that


λ → 0

lim

N

i = 1

∑ γ t i − t i − 1 , x = 0

Both the ordinary integrals and the stochastic Ito integrals are contained in the sum underconsideration. Ordinary integrals are estimated by standard methods. So, as an example, weshall estimate one of the sums of stochastic integrals

E |

N

i = 1

∑t i

t i − 1

∫ C r, x ∇V r, x − C ti − 1, x ∇V ti − 1, x d w r |2 ≤

≤ t | t − t | < λ

max |C t

, x − C t, x |

2max

xE | ∇ V t , x |

2+

+t

max |C t , x |2

| t − t | < λ

max E | V t

, x − V t |

2

From the condition of theorem 1.7 it follows that the expression inbrackets tends to 0 asλ → 0 . Passing to the limit in equality ( 1. 12 ) as and supposingk = N we can verifythat the functionv t, x = v t , x ; 0, ϕ being a solution of equation (1.6) is also asolution of equation (1.4). Thus by conditions of theorem 1.7 each ”classical” solution of theCauchy problem (1.6) is also a solution of the equation (1.4) and vice versa. By virtue oftheorem 1.2 the solution of the problem (1.4) is unique and consequently thesolution of theproblem (1.6) is also unique. That theorem is now proved.

We return now to the problem ( 1.9 ). Let us recall that functionv t , x belonging to

the spaceB 2

2 + γmeasurable with the flow ofσ −algebras Ϝ 0 , t

1 independent of the

Wiener processesw j t , j = 2, 3, ..., l contained in the system (1.8) and satisfyingthe equality (1.9) for allt and x at once with probability is called the ”classical” solution ofthe problem (1.9). It is rather difficult to apply the methods analogous tothose in the proof ofTheorem 1.7 directly. Therefore another technique of proving the uniqueness of the solution ofequation (1.9) will be used.

Theorem 1.8. Let the conditions of the theorem 1.6 hold and assume that the functionsb t , x , C i t , x i = 1, 2, ..l for any t ∈ 0 , T are equal 0 outsidesome compact. Then the ’classical ’ solution of the problem (1.9) is unique.

Proof. Suppose equation ( 1.9 ) has two ”classical” solutionsv i t , x i = 1, 2.Denote their difference byh t , x = h 1 t , x , ...,h n t , x . Then


h q t , x =t

0

∫ m

i = 1

∑ b i s , x ∂ h q s , x

∂ x i

+

+m

i , j = 1

∑I

k = 1

∑d k

ν = 1

∑ c i , ν k

s, x c j , ν k

s, x ∂ 2

h s, x

∂ x i ∂ x j

+ f q s, x, v 1 s, x −

− f q s, x, v2 s, x d s +m

i = 1

∑d 1

ν = 1

∑t

0

∫ c i , ν 1

s, x ∂ hq s, x

∂ x i

d w ν 1

s

q = 1, 2, ..., m . Treating the variablex in the preceding equality as a parameter, we

apply Ito’s formula to the functionh q

2 s , x . Then

h q

2 s, x =

t

0

∫ 2 h q s, x m

i = 1

∑ b i s, x ∂ h q s, x

∂ x i

+

+m

i , j = 1

∑I

k = 1

∑d k

ν = 1

∑ c i , ν k

s, x c j , ν k

s, x ∂ 2

h s, x

∂ x i ∂ x j

+

+ f q s, x, v 1 s , x − f q s, x, v 2 s, x d s +

+t

0

∫m

i , j = 1

∑d 1

ν = 1

∑ c i , ν 1

s, x c j , ν 1

s, x ∂ h q s, x

∂ x i

∂ h q s, x

∂ x j

d s +

+m

i = 1

∑d 1

ν = 1

∑t

0

∫ c i , ν 1

s , x ∂ h q

2 s , x

∂ x i

d w ν 1

s

We shall now take the mathematical expectation from the equality, and integrate the

equality obtained with respect tod x . Note, that according to the assumption of the theorem,the order of integration can be changed. Indeed, for the Riemann integral it is sufficient, thatintegrand function be absolutely integrable with respect to measuredx × dt with the


probability 1. For the order of integration to be reversed in the stochastic integral it is sufficient[39] , that with probability1

T

0

∫E m

∫ | C 1

t , x ∇ h2 t , x d x d t < ∞ #

Since the functionh ∗ belongs to the spaceB 2

2 + γ, and the diffusion coefficient

C 1

∗ is equal to 0 outside some compact , then condition (1.14) holds. Integratingby parts in the inner integral in the first two terms on the right-hands side, we obtain

E m

∫ E h q

2 s , x d x =

t

0

∫ d s

E m

∫ m

i = 1

∑ ∂ b i s , x

∂ x i

+

+m

i , j = 1

∑I

k = 1

∑d k

ν = 1

∑∂ 2

c i , ν k

s , x c j , ν k

s , x

∂ x i ∂ x j

E h q

2 s, x + #

+m

i , j = 1

∑I

k = 1

∑d k

ν = 1

∑ c i , ν k

s, x c s, x ∂ h q

2 s, x

∂ x i

∂ h q

2 s, x

∂ x j

+

+m

i , j = 1

∑d 1

ν = 1

∑ c i , ν 1

s, x c j , ν 1

s, x ∂ h q

2 s, x

∂ x i

∂ h q s, x

∂ x ij

+

+ k f E h q

2 s , x d x .

Here,k f = sup t , x , v | ∇ v f t , x , v | . Summing over allq we establish

estimates for the functionh t , x = v 1 t , x − v 2 t , x :

E m

∫ E h q

2 t , x d x ≤

t

0

∫ d s

E m

∫ div b s , x +

+ 12

∇ , ∇ C s, x C∗ s, x + k f n E | h s , x | d s

By Granules’s lemma,


E m

∫ E | v 1 t , x − v 2 t , x |2

d x = 0.

Since subintegral function is continuous, the solutions coincide for all t , x at once withprobability 1, which is what had to be proved.

Now we are going to study other types of stochastic dynamic system.Note, that one of thepossible applications of the theory introduced above is design of the methods of resolving theproblem Cauchy for nonlinear stochastic parabolic equation. Turn back to the Theorem 1.5 wewould to emphasize that there are two types of ’white’ noise terms involved into the equations.They are the terms which we can conditionally call ’external’ and ’internal’. Internal noise isgenerated by the diffusion of the processξ ∗ . External noises in the equation ( 1.9 ) are’white’ noise terms that perturb the second equation of the system (1.8) . Analyzing equations(1.4), (1.8 ) we can see that the macro-parametersu t , x or v t , x change alongthe trajectories of the characteristicsξ ∗ . Now we represent a scheme where theseparameters will changed along the distribution of the characteristics. This is the directextension of Kolmogorov’s equations and can be considered as a generalization of probabilisticmethods of studying deterministic nonlinear parabolic equations [ 4]. Given the conditions ofthe theorem (1.5) we introduce the problem

v t, x = E ϕ ξ 0 ; t, x +

+t

0

∫ E f r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d r +

+t

0

∫ E g r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d w r #

The major deference between equations (1.16) and (1.4) is that the function v ∗ in(1.16) depends upon the averaged characteristics and the functionu ∗ in the (1.4)depends upon the characteristics itself. The proof of the existencetheorem is almost the sameas before. Some insignificant modifications involved based on the new term in (1.16). To usecontraction mapping principle we need to substitute random functions

v , , ξ ∗ ; t, x into conditional expectation.We will show that after substitution correspondent stochastic integral remain measurable

function. First, letB be a set of Borel’s bounded random functions depending ont , x andv t , x, ω be an arbitrary function belonging toB . FunctionG s, t, x, ω = g r , ξ r ; t, x , v r , ξ r ; t, x is measurable overs, t, x, ω as a composition of measurable functions and


s , t , x

sup E | G s, t, x, ω | < ∞

Now , we can exhibit that there exist measurable modification of theconditional

expectationE G s, t, x, ω | Ϝ s .Lemma 1.2.Let G s, t, x, ω be a measurable random field ,and

sup s , t , x E | G s, t, x, ω | < ∞. Then there exists , t , x ω − measurable

modification of the conditional expectationE G s, t, x, ω / Ϝ s .Proof. DenoteU a set of random functions for which Lemma is true. LetA be a

s, t, x, ω − measurable set of a typeA = A 1 × A 2 × A 3 × A 4 , whereA 1 , A 2 , A 3 are Borel sets from 0 , t , 0 ,T , E

mrespectively, and

A 4 ∈ Ϝ . Then

E χ A s, t, x, ω / Ϝs = E χ A 1 s χ A 2

t χ A 3 x χ A 4

ω / Ϝs =

= χ A 1 s χ A 2

t χ A 3 x E χ A 4

ω / Ϝ s

From general martingale theory [35] it follows that there exist measurable modification ofright hand side of the equality. SetU is an algebra and monotone class and contains indicatorsof the measurable sets of chosen type. Hence,U contains indicators of alls, t, x, ω − measurable sets. Evidently, thatU is linear and closed with respect to

monotone limit transition. So thatU contains all measurable nonnegative functions. Since, anymeasurable function can be represented as a difference of two nonnegative functions one gets aproof of the lemma.

Turn back to the problem (1.16). The solution of the problem exist, unique and satisfiesrelationship

v t, x = E v ξ s ; t, x / Ϝ s +

+t

s

∫ E f r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d r + #

+t

s

∫ E g r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d w r

We will establish a derivation of the first term in right hand sideof (1.17). Two othersterms can be establish without any difficulties. Since, the randomvariableξ s ; t, x is

independent onϜ s then


E E ϕ ξ 0 ; t, y | y = ξ s ; t, x =

= E E ϕ ξ 0 ; t, x | y = ξ s ; t, x / Ϝ s

Taking into account that the right hand side of (1.16) ist, x, ω − measurable note

E ϕ ξ 0 ; t, x +

0

s


+s

0

∫ E g r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d w r =

= E E ϕ ξ 0 ; s, y +

+s

0

∫ E f r, ξ r ; s, y , v r, ξ r ; s, y / Ϝr d r +

+s

0

∫ Eg r, ξ r; s, y , v r, ξ r ; s, y / Ϝr dw r y = ξ s; t, x / Ϝs

= E v ξ s ; t, x / Ϝ s

It is not difficult to check that the statements of the theorems1.2 - 1.4 remain correct for the

system (1.16). Granting these we are able to formulateTheorem 1.9.Under conditions of the theorem 1.5 the solution of the equation (1.16) is

a solution of the Cauchy problem

v t , x = ϕ x +t

0

∫ m

i = 1

∑ ∂ v s , x

∂ x i

b i s , x +

+ 12

i , j = 1

m

∑ ∂ 2v s, x

∂ x i ∂ x jk = 1

d

∑ c i k s, x c j k s, x + f s, x, v s, x d s +


+t

0

∫ g s, x, v s, x d w s

The proof of the theorem is almost the same as the theorem 1.5 . We just have to start fromequality

v t, x − v s, x = E v ξ s ; t, x − v s, x / Ϝ s +

+t

s


+t

s

∫ E g r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r d w r

and then to repeat the way of the proof of the theorem 1.5.

Quasilinear evolutionary stochastic systems

In this section we will consider quasilinear stochastic parabolic systems. Two significantdifferences between a quasilinear and a semilinear systems should be noted. The firstdifference is that the solution of quasilinear systems is local in time and the solution ofsemilinear systems exists on an arbitrary finite interval of time. The second one is that thesolution of quasilinear systems with smooth coefficients is continuous in x and therefore thesolution must be interpreted as generalized. For semilinear systems differentiability of thesolution inx corresponds to the differentiability of the coefficients inx . Some types ofstochastic partial differential equations were studied in [12,36,41].

We introduce the needed notation. Let thew j t be mutually independent Weiner

processes with values ind j -dimensional Euclidean spaceEd j , d j ≥ 1 , j = 1, 2 ,

and let w t = w 1 t , w 2 t . For an arbitrary intervalΔ ⊂ 0 , T let

Ϝ Δ j

= σ w j t 1 − w j t 2 ; t 1, t 2 ∈ Δ and Ϝ Δ =2

j = 1

∪ Ϝ Δ j

.

Consider the quasilinear system


ξ s ; t, x = x +t

s

∫ b r, ξ r ; t, x , v r, ξ r ; t, x d r −

−t

s

∫ E c r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r , t d←w r #

u t, x = ϕ ξ 0 ; t, x +t

0

∫ f r, ξ r ; t, x , v r, ξ r ; t, x d r +

+t

0

∫ g r , ξ r ; t, x d←w 1 r

where v t , x = E u t , x / Ϝ 0 , t

1 and 0 ≤ s ≤ t ≤ T . The arrow

here over the Wiener processw t means that the stochastic integral is to be interpreted asIto’s inverse integral . A solution of the system (2.1) on the interval 0 , T is understoodto be a pair of random functionsξ s ; t, x , u t , x that are defined fors ≤ t < T and satisfy the estimate (2.2), are measurable in all their arguments, aremeasurable with the respective flows ofσ −algebrasϜ s , t and Ϝ 0 , t , and satisfy (2.1)

for all s, t, and x with probability 1. In what follows to simplify the notation the exact samesymbol | ∗ | will be used for the norm of a vector in Euclidean space and for the norm of thetrace of a matrix.

THEOREM 2.1Suppose that the nonrandom functionsϕ x , f t , x , u withvalues inE

nand Borel measurable in all their arguments, the functionb t , x , u with

values in Em

, the matrix-functiong t , x of dimensiond 1 × n and matrix- functionc t , x , u = c 1 t , x , u , c 2 t , x , u , where c j t , x , u is ofdimensiond j × m, are uniformly bounded by a positive constantK and satisfy a Lipschitzcondition inx andu with a positive constantL. Then there exist a time interval 0 , T inwhich the system (2.1) has unique solutionu t , x , ξ s ; t , x in the class of thefunctions

x , t 1 ≠ t 2

sup E |ξ s ; t 1 , x − ξ s ; t 2 , x

| t 1 − t 2 |0 . 5

| p+

+x , s 1 ≠ s 2

sup E |ξ s 1 ; t , x − ξ s 2 ; t , x

| s 1 − s 2 |0 . 5

|p

+


+x

sup E | u t, x |p+

x ≠ y

sup E |u t , x − u t , y

x − y|

p+ #

s , x

sup E |ξ s ; t, x

1 + | x ||

p+

s , x ≠ y

sup E |ξ s ; t, x − ξ s ; t, y

x − y|

p+

+t ≠ s , x

sup E |u t , x − u s , x

| t − s |0 . 5

|p

< ∞

for any p ≥ 2 , and 0 ≤ s ≤ t < T . In addition, a solution to system (2.1) is uniquein the class of function that satisfy ( 2.2 ).

Proof. Before proceeding to prove the theorem , we point out that the conditionalexpectation in the integrand of stochastic integral has a predictable modification. It can beproved the same way as Lemma 2 . Rewriting the system (2.1) in operator form

ξ s ; t , x = L s ; t , x ξ , u

u t , x = L t , x ξ , u

we form successive approximations

ξ 0

s ; t , x = x ,

ξ n

s ; t , x = L s ; t , x ξ n − 1

, u n

,

u n

t , x = L t , x ξ n − 1

, u n

.

This choice of the successive approximations give us possibility to use the results of the

seilinear case that considered above. The first approximationξ 0

s ; t, x = x,

determines a functionu 1

t , x that satisfies (2.2) in an arbitrary interval of 0 , ∞ .

Suppose now that the functionsξ n

s ; t , x , u n

t , x were found and letL n denote the constant on right -hand of (2.2). Show that there is a constant L n + 1 that

majorized the left-hand side of (2.2), where

ξ s ; t, x = ξ n

s ; t, x , u t, x = u n

t, x

This shows that the procedure of construction successive approximations does not


terminate. We now indicate the time-interval in which the sequence of numbersL n isuniformly bounded. LetΔ be an arbitrary interval of timep ≥ 2 , and R t , ξ , u bethe expression of the right hand side of ( 2.2 ). By definition we put

R n =t ∈ Δ

sup R t , ξ n

, u n

We will establish the constantR n + 1 for which

R n + 1 =t ∈ Δ

sup R t , ξ n + 1

, u n + 1

We have

1 + | x |p

− 1E | ξ

n s ; t , x |

p ≤ 3p − 1

1 + Kpα p t

E | u n + 1

t , x |p ≤ 3

p − 1K

p 1 + α p t

where α p t = p2 p − 1 t

p2 − 1

+ tp − 1

. Denote

u t

n

p =x ≠ y

sup E |u t , x − u t , y

x − y|

p,

ξ s , t

n

p =x ≠ y

sup E |ξ s ; t , x − ξ s ; t , y

x − y|

p

Then

u t

n + 1

p ≤ 3p − 1

Lp ξ 0 , t

n

p +

+ α p t t

0

∫ 1 + u s

n

p ξ 0 , t

n

p d s

With Gronwall’s lemma, we obtain


t ≤ T

sup u t

n + 1

p ≤ U n , p

where

U n , p = 3p − 1

L R np 1 + T α p T exp 3

p − 1α T L R n

pT

Similarly ,

ξ s , t

n

p ≤ V n , p ,

t ≠ s , x

sup E |u

n + 1 t, x − u

n + 1 s, x

| t − s |0 . 5

|p ≤ 3

p − 1 L R n

p×

× 1 + 2p − 1

T α p T 1 + U n , p + 2p − 1

T α p T ,

x , t 1 ≠ t 2

sup E |ξ

n + 1 s ; t 1 , x − ξ

n + 1 s ; t 2 , x

| t 1 − t 2 |0 . 5

|p ≤

≤ 43

p − 1

V n , p + Kp

Tp

+ α p T ,

x , s 1 ≠ s 2

sup E |ξ

n + 1 s 1; t, x − ξ

n + 1 s 2 ; t, x

| s 1 − s 2 |0 . 5

|p ≤

≤ 2p − 1

Kp

1 + α p T

where

V n , p = 3p − 1

1 + T α p T L R np 1 + U n , p

One can point out an interval of time where numbersV n , p , U n , p are uniformly


bounded. We have

E | u n + 1

t , x − u n + 1

t , y |p ≤

≤ β p t s

sup E | ξ n

s ; t, x − ξ n

s ; t, y |p

;

ssup E | ξ

n s ; t, x − ξ

n s ; t, y |

p ≤ 3p − 1

| x − y |p

+

+ μ p t s

sup E | ξ n − 1

s ; t, x − ξ n − 1

s ; t, y |p

where

β p t = 3p − 1

Lp 1 + t 1 + α p t exp 3

p − 1L

pt α p t ,

μ p t = t 1 + α p t 1 + β p t Lp

The functionμ p t is continuous and monotone increasing andμ p 0 = 0. Let

t 1 denote the root of equationμ p t = 31 − p

. Then, for any t < t 1

ssup E | ξ

n s ; t, x − ξ

n s ; t, y |

p ≤

≤3

p − 1

1 − 3p − 1

μ p t | x − y |

p #

E | u n

t, x − u n

t, y |p ≤

3p − 1

β p t

1 − 3p − 1

μ p t | x − y |

p

Using these estimates, we can prove that the successive approximations converge. Let

B p

t, N p

tbe Banach spaces of the random functions

Ϝ t , Ϝt

, t ∈ 0 , T with norms


|| u || t , p

p=

x , s ≤ t

sup E | u s , x |p

,

||| ξ ||| t , p

p=

x , s ≤ t

sup E |ξ s ; t , x

1 + | x ||

p,

respectively. Putting for simplicityp = 2 we have

|| u n + 1 − u

n || t , 2

2 ≤ 3 L2 1 + t α 2 t +

+ 2α 2 t t

0

∫ || u n + 1 − u

n || s , 2

2d s + 2α 2 t ×

×t

0

∫x

sup EE | u n + 1

s, z 1 − u n + 1

s, z 2 |2

z 1 = ξ n s ; t , x

z 2 = ξ n − 1 s ; t , x

d s

Applying the formulas (2.3) we find that

t

0

∫x

sup E E | u n + 1

s, z 1 − u n + 1

s, z 2 |2z 1 = ξ n s ; t , x

z 2 = ξ n − 1 s ; t , x

d s ≤

≤3 β 2 t

1 − 3 μ 2 t ||| ξ

n − ξ n − 1

||| t , 2

2

Applying Gronwall’s lemma, we obtain

|| u n + 1 − u

n || t , 2

2 ≤ γ t ||| ξ n − ξ

n − 1 ||| t , 2

2

where γ t = 3 L2

1 + t α 2 t + 6 t β 2 t

1 − 3 μ 2 t exp 6 L

2t α 2 t .

Similarly to the preceding

||| ξ n − ξ

n − 1 ||| t , 2

2 ≤ 2 L2α 2 t t ||| ξ

n − 1 − ξ n − 2

||| t , 2

2+


+ 2t || u n − u

n − 1 || t , 2

2+ 2β 2 t

t

0

∫x

sup E E | ξ n − 2

s ; t, z 1 −

− ξ n − 2

s ; t, z 2 |2

/z 1 = ξ n − 1 s ; t , x , z 2 = ξ n − 2 s ; t , x

d s ≤

≤ λ t ||| ξ n − 1 − ξ

n − 2 ||| t , 2

2

where λ t = 2 L2α 2 t

1 + 2 γ t + 6 β 2 t

1 − 3 μ 2 t . We point out that

λ t is continuous and monotone increasing, andλ 0 = 0 . Thus, λ t 2 = 1 for some positivet 2 and therefore

||| ξ n − ξ

n − 1 ||| t , 2

2 ≤ λn t ||| ξ

1 − ξ 0

||| t , 2

2

|| u n − u

n − 1 || t , 2

2 ≤ γ t λn t ||| ξ

1 − ξ 0

||| t , 2

2

Let T = min t 1 , t 2 . Then, for t < T , the preceding inequalities lead to

n , m → ∞

lim || u n − u

m ||| t , 2

2+ ||| ξ

n − ξ m

||| t , 2

2 = 0

This means that there exist processesξ s ; t , x , u t , x for which

m → ∞lim || u − u

m ||| t , 2

2+ ||| ξ − ξ

m ||| t , 2

2 = 0

when t < T .It easy to verify that

E | ξ n

s 1 ; t , x − ξ n

s 2 ; t , x |p ≤

≤ 2p

Kp 1 + α p t | s 1 − s 2 |

p2 ,


E | ξ n

s ; t 1 , x − ξ n

s ; t 2 , x |p ≤ k

1 t

∗ | t 1 − t 2 |

p2 ,

E | u n

t 1 , x − u n

t , x |p ≤ k

2 t

∗ | t 1 − t 2 |

p2 ,

where t∗

= max t 1 , t 2 and

k 1

t = 4p − 1

Kp 1 + α p t exp 4

p − 1L

p 1 + α p t t ;

k 2

t = 6p − 1

Kp 1 + α p t + 3

p − 1L

p 1 +

+ 2p − 1

1 + α p t 1 +3 β p t

1 − 3 μ p t k

1 t .

Letting n → ∞ in formula (2.3) we can show that the processes

ξ s ; t , x , u t , x belong to the spaces Bpt

, N p

t, satisfy (2.2) and therefore

have measurable separable modification. Retain the same notation for them as before. Now,we are able to show that this modifications are solutions of the system(2.1). To this end itsuffices to show possibility to pass to the limit in each of the terms in the system of successiveapproximations. Justifying the passage to the limit in each of the terms occurring in the systemis basically the same and so we shall explain it just for one of the stochastic integrals. Applyingthe properties of conditional expectations and using (2.2) fort < T we note

E |t

s

∫ E c r, ξ n − 1

r ; t, x , v r , ξ n − 1

r ; t, x −

− c r , ξ r ; t , x , v r , ξ r ; t , x / Ϝr d w← r | 2 ≤

≤ 2 L2

t

s

∫ E | ξ n − 1

r ; t, x − ξ r ; t, x |2 1 + || v

n || r

2 +

+ || u n − u || r

2 d r ≤ N 1 ||| ξ

n − 1 − ξ ||| t

2+ || u

n − u || t

2


In what follows, N i = N i t , i = 1 , 2 , ... will denote continuous increasingfunctions on 0 , T . By means of analogous reasoning , by passing to the limit in thesystem of successive approximations asn → ∞ we can show that the functionsξ s ; t , x , u t , x give a solution to the system (2.1) on 0 , T for which the

inequality (2.2) holds. To prove uniqueness of the solution let suppose thatξ j s ; t , x and u j t , x , j = 1, 2 be two solutions of the system (2.1)satisfying (2.2) and specified on the same time interval. Then

E | ξ 1 s ; t, x − ξ 2 s ; t, x |2 ≤ 2 1 + t L

2×

×t

s

∫ E | ξ 1 r ; t, x − ξ 2 r ; t, x |2

+

+ | u 1 r, ξ 1 r ; t, x − u 2 r, ξ 2 r ; t, x |2 d r ≤

≤ N 2

t

s

∫ E | ξ 1 r ; t, x − ξ 2 r ; t, x |2

+

+x

sup E | u 1 r, x − u 2 r, x |2 d r .

By Grownwall’s lemma , we find that

E | ξ 1 s ; t, x − ξ 2 s ; t, x |2 ≤

≤ N 2

t

s

∫x

sup E | u 1 r, x − u 2 r, x |2

d r .

In exactly the same way one can establish

xsup E | u 1 t, x − u 2 t, x |

2 ≤ N 4 ||| ξ 1 − ξ 2 ||| t

2.

Taking these inequalities into account, we have

xsup E | u 1 t, x − u 2 t, x |

2 ≤ .


≤ N 5

t

s

∫x

sup E | u 1 r, x − u 2 r, x |2

d r

From this, we easily deduce that

xsup E | u 1 t , x − u 2 t , x |

2= 0 ,

and hence

s , x

sup E | ξ 1 s ; t , x − ξ 2 s ; t , x |2

= 0 .

Theorem is proved.In what follows, it will be shown that the functionv t , x determined by the system

(1) form generalized solution of a stochastic Cauchy problem for a class of quasilinearsystems. First we find estimates for the derivatives ofu t , x The system determining

ξ 1

s ; t , x = ∇ ξ s ; t , x ,

u 1

t , x = ∇ u t , x ,

where ∇ = ∂

∂ x 1

, .... ∂

∂ x m

, which is founded by formally differentiating the

original one, has more complicated form and we have not succeededin proving its solvabilityby the same method as Theorem 2 1. More precisely, the right-hand side of the system

contains an integral of the product ofu 1

and ξ 1

which is not allowed by thehypotheses of the theorem 2.1. Therefore when speaking of estimates of derivatives of thesolution of the system (2.1) in what follows, we shall always bear in mind their a priori nature.

Nevertheless, it should be noted that if the first derivativesξ 1

s ; t , x or

u 1

t , x have been shown to exist , then proving the existence of the high orderderivatives is of no difficulty, since the equations for them satisfy the hypotheses of Theorem2.1 and the proof is similar to the theorem on the continuous dependence of the solution to thesystem (2.1) on the coefficients. Formula (2.2) may be used to obtain a priori estimates for

ξ 1

s ; t , x and u 1

t , x . However, it is inadequate for obtaining estimates forthe higher derivatives. Assume for arbitraryp ≥ 2 that there exists a function

u 1

t , x such that

h → 0

limx

sup E |u t , x + h − u t , x

h− u

1 t , x |

p= 0

for anyt in some subinterval of 0 , T introduced in Theorem 2.1. All of the


subsequence discussion will be carried out on this subinterval.

THEOREM 2.2Assume that the functions

ϕ n x , b n t , x , u , c n t , x , u , f n t , x, u , g n t , x satisfy the conditions of theorem 2.1 and

n → ∞

limx

sup |ϕ n x − ϕ x | +T

0

∫x , u

sup | b n t, x, u − b t, x, u | +

+ | c n t, x, u − c t, x, u | +

+ | g n t, x − g t, x | + | f n t, x, u − f t, x, u | d t = 0

Then , solutionsu n t, x , ξ n s ; t, x converge tou t, x , ξ s ; t, x

n → ∞lim

t , x

sup E | u n t, x − u t, x |2+ | ξ n s ; t, x − ξ s ; t, x |

2 = 0

Here u t, x , ξ s ; t , x is the solution of the system ( 2.1 ).Proof. The proof of this theorem is similar to correspondent proof of the theorem 1.2 and

so we just briefly remind the main stand points. One has

s , x

sup E | ξn s ; t, x − ξ s ; t, x |2 ≤

≤ 8t

0

∫u , x

sup t | b n r, x, u − b r, x, u |2+

+ | c n t, x, u − c t, x, u |2 d s + 4 L

2 1 + t 1 + 2 || u || t

2 ×

×t

0

∫x

sup E | u n r, x − u r, x |2+ | ξ n r ; t, x − ξ r ; t, x |

2 d r ;


t , x

sup E | u n t, x − u t, x |2 ≤ 12

t

0

∫u , x

sup | g n s, x − g s, x |2+

+ | f n s, x, u − f s, x, u |2 d s + 3

x

sup |ϕ n x − ϕ x |2+

+ 12 L2 1 + t 1 + 2 || u || t

2

t

0

∫x

sup E | u n r, x − u r, x |2+

+ | ξ n r ; t, x − ξ r ; t, x |2 d r .

Adding together these two inequalities and applying Grownwall’s lemma , we get the

proof. Consider the equation

η s ; t, x = I +t

s

∫ b 1

r ; t, x, η r ; t, x d r − #

−t

s

∫ E c 1

r ; t, x, η r ; t, x / Ϝ d←w r

where

b 1

r ; t, x, η r ; t, x = ∂ b

∂ x r, ξ r ; t, x , v r, ξ r ; t, x +

+∂ b

∂ u r, ξ r ; t, x , v r, ξ r ; t, x v 1 r, ξ r ; t, x η r ; t, x ;

c 1

r ; t, x, η r ; t, x = ∂ c

∂ x r, ξ r ; t, x , v r, ξ r ; t, x +

+∂ c

∂ u r, ξ r ; t, x , v r, ξ r ; t, x v 1 r, ξ r ; t, x η r ; t, x ;


v 1

t , x = E u t , x | Ϝ 0 , t

1

The system (2.4) is derived from the equation for characteristic occurring in (2.1) by

differentiating with respect to parameterx .Lemma 2.1.Suppose that the hypotheses of Theorem 2.1 hold and that the functions

b t, x, u and c t, x, u have bounded continuous first derivatives inx andu . Then

the system (2.4) has a unique solution withη r ; t, x = = ξ 1

r ; t, x , andthe derivative is understood in the sense of mean square.

Proof.Since the proof of the lemma is similar to the corresponding one for ordinarystochastic differential equations, we shall state only main steps. Denote the right hand side of(2.4) by

L η s ; t, x . Then

E | Lη s ; t, x |2 ≤ 3 1 + 1 + T K

2×

× || 1 + u 1

|| t

2t

s

∫ E | η r ; t, x |2

d r ;

E | L η s ; t, x − L ζ s ; t, x |2 ≤

≤ 2 1 + T K2|| 1 + u

1 || t

2t

s

∫ E | η r ; t, x − ζ r ; t, x |2

d r

From this inequalities, it is easy to see that some power of operator L is a contractingoperator in the Banach space of random functions with finite second order moments. Thisimplies the first assertion of the lemma. We then have

E | η s ; t, x −ξ s ; t, x + Δ x − ξ s ; t, x

Δ x|

2 ≤

≤ 2t

s

∫ E t | B 1

r ; t, x − B Δ 1

r ; t, x |2

+


+ | C 1

r ; t, x − C Δ 1

r ; t, x |2 d r

To simplify notation we putB 1

r ; t, x = b 1

r ; t, x, η r ; t, x , and

we have taken forB Δ 1

r ; t, x a similar expression corresponding to the driftcoefficient of the process

ξ s ; t, x + Δ x − ξ s ; t, x

Δ x

and C 1

r ; t, x , C Δ 1

r ; t, x are given similarly. More precisely,

B Δ 1

r ; t, x = Δx− 1

b r, ξ r ; t, x + Δx , v r, ξ r ; t, x + Δx −

− b r , ξ r ; t, x , v r , ξ r ; t, x

C Δ 1

r ; t, x = Δx− 1

c r, ξ r ; t, x + Δx , v r, ξ r ; t, x + Δx −

− c r, ξ r ; t, x , v r, ξ r ; t, x

Since the estimation of both terms in the integrand is identical, we shall do one of them.We find that

E | B 1

r ; t, x − B Δ 1

r ; t, x |2 ≤

≤ 4 E | ∇ x b r , ξ r ; t, x , v r, ξ r ; t, x −

−1

0

∫ 1 − θ ∇xb r, ξθ r ; t, x , v r, ξ r ; t, x + Δ x d θ |2||| η ||| t

2+

+ 4 K2

E | η s ; t, x −ξ s ; t, x + Δ x − ξ s ; t, x

Δ x|

2+

+ 6 E | ∇ u b r, ξ r ; t, x , v r, ξ r ; t, x −


−1

0

∫ 1 − θ ∇ u b r , ξ r ; t, x , v r , ξ θ r ; t, x d θ |2×

× | η r ; t, x |2

| v θ r , ξ r ; t, x |2

+

+ 6 K2E || u 1 || t

2E | η s ; t, x −

ξ s ; t, x + Δ x − ξ s ; t, x

Δ x|

2+

+ 6 K2E | ξ

1 r ; t, x |

2| v

1 r , ξ r ; t, x −

−t

0

∫ 1 − θ v 1

r , ξ θ r ; t, x d θ |2

whereξ θ r ; t, x = θ ξ r ; t, x + 1 − θ ξ r ; t, x + Δ x . Applying

Holder’s inequality and using the conditions of the Lemma and finiteness of the moments ofthe random functions we easily find that

x

sup E | η s ; t, x −ξ s ; t, x + Δ x − ξ s ; t, x

Δ x|

2 ≤ Δ x +

+ R

0

t

∫x

sup E | η r ; t, x −ξ r ; t, x + Δ x − ξ r ; t, x

Δ x|

2d r

whereΔ x → 0

lim Δ x = 0 and R = R || v 1

|| , K . The proof of Lemma 2.1

is easily completed by resorting to Gronwall’s lemma.

Remark. Taking lemma 2.1 as our starting assumption and applying the theorem oncontinuous dependence of a solution on the coefficients with some insignificant complications,

we can prove that the processu 1

t , x satisfies the equation obtained from (2.1) bydifferentiating formally with respect tox . Therefore the system

u 1

t, x = Φ 1

0 ; t, x +t

0

∫ F 1

r ; t, x d r +


+t

0

∫ G 1

r ; t, x d←w 1 r #

ξ 1

s ; t, x = I +t

s

∫ B 1

r ; t, x d r +

+t

s

∫ E C 1

r ; t, x / Ϝ 0 , t d←w r

will be considered as we investigate further the a priori smoothness of the function

u t , x . Here Φ 1

0 ; t , x = ∇ ϕ ξ 0 ; t, x ξ 1

0 ; t, x and

F 1

r ; t , x and G 1

r ; t , x are given by expressions similar to

B 1

r ; t , x in Lemma 2.1.LEMMA 2.2. Under hypotheses of Lemma2.1, suppose that the partial derivatives ofthe

coefficients of the system (2.1) satisfy a Holder condition inx and u with exponentγ ∈ 0 , 1 . Then, for p ≥ 2

x ≠ y

sup E |u

1 t, x − u

1 t, y

| x − y |γ |

p+

+ |ξ

1 s ; t, x − ξ

1 s ; t, y

| x − y |γ |

p < ∞

Proof. We have

E | ξ 1

s ; t , x − ξ 1

s ; t , y |p ≤

≤ 2p − 1

s

t

∫ E tp − 1

| B 1

r ; t, x − B Δ 1

r ; t, x |p+

+ p2

p − 1 t12 p − 1

| C 1

r ; t, x − C Δ 1

r ; t, x |p d r ;

E | u 1

t, x − u 1

t, y |p ≤


≤ 3p − 1

E | Φ 1

0 ; t, x − Φ 1

0 ; t, y |p

+

+ ∫ s

tE t

p − 1| F

1 r ; t, x − F Δ

1 r ; t, x |

p+

+ p2

p − 1 t12 p − 1

| G 1

r ; t, x − G Δ 1

r ; t, x |p

d r .

As an example, we shall estimate one of the terms on the right-hand side:

t

0

∫ E | F 1

r ; t, x − F Δ 1

r ; t, x |p ≤

≤ 2p − 1

E

t

0

∫ | ∇ x f r, ξ r ; t, x , v r, ξ r ; t, x ξ 1

r ; t, x −

− ∇ x f r, ξ r ; t, y , v r, ξ r ; t, y ξ 1

r ; t, y |p d r +

+ | ∇u f r, ξ r ; t, x , v r, ξ r ; t, x v 1 r, ξ r ; t, x ξ 1 r ; t, x −

− ∇ u f r, ξ r ; t, y , v r, ξ r ; t, y v 1

r, ξ r ; t, y ×

× ξ 1

r ; t, y |p d r = 2

p − 1E

t

0

∫ I 1 r + I 2 r d r .

In the subsequent computations,N i , i = 1,2... , denote positive constants notdepending onx or y . Applying Hölder inequality, we easily obtain the estimate

I 2 r ≤ N 1 E | ξ 1

r ; t , x − ξ 1

r ; t , y |p

+

+ N 2 E | u 1

r , x − u 1

r , y |p

+ N 3 | x − y |p γ

.

A similar estimate also holds forI 1 r

I 1 r ≤ N 5 E | ξ 1

r ; t, x − ξ 1

r ; t, y |p+ N 4 | x − y |

p γ.


Since

E | Φ 1

0 ; t, x − Φ 1

0 ; t, y |p ≤

≤ 2p − 1

E G γp

| ξ 1

r ; t, x − ξ 1

r ; t, y |p γ

+

+ Kp

| ξ 1

r ; t , x − ξ 1

r ; t , y |p,

it is not hard to see that

+ |ξ

1 s ; t , x − ξ

1 s ; t , y

| x − y |γ |

p ≤ N 6 +

+ N 7

t

0

∫x ≠ y

sup E |u

1 r , x − u

1 r , y

| x − y |γ |

p+

+ |ξ

1 r ; t , x − ξ

1 r ; t , y

| x − y |γ |

p d r

Applying Gronwall’s lemma, we see that Lemma 2.2 is true.

Define expressionsB 2

r ; t, x , C 2

r ; t, x , Φ 2

r ; t, x ,

F 2

r ; t, x , G 2

r ; t, x by formal differentiation with respect tox of the

correspondent coefficients of the system ( 2.1 ). For instance

Φ 2

0 ; t, x =

= Tr ξ 1

0 ; t, x ∗ ∂ 2

ϕ ξ 0 ; t, x

∂ x2 ξ

1 0 ; t, x +

+∂ ϕ ξ 0 ; t , x

∂ xξ

2 0 ; t , x ,


B 2

r ; t, x = Tr ξ 1

r ; t, x ∗ ∂ 2

b ξ r ; t, x

∂ x2 ξ

1 r ; t, x +

+ v 1

r , ξ r ; t, x ξ 1

r ; t, x ∗×

×∂ 2

b ξ r ; t , x

∂ x ∂ uξ

1 r ; t, x + ξ

1 r ; t, x

∗×

×∂ 2

b ξ r ; t, x

∂ x ∂ uv

1 r, ξ r ; t, x ξ

1 r ; t, x +

v 1

r, ξ r ; t, x ξ 1

r ; t, x ∗ ∂ 2

b ξ r ; t, x

∂ u2 ×

× v 1

r, ξ r ; t, x ξ 1

r ; t, x +

+∂ b ξ r ; t, x

∂ xξ

2 r ; t, x +

∂ b ξ r ; t, x

∂ u×

× ξ 1

r ; t, x ∗

v 2

r , ξ r ; t, x ξ 1

r ; t, x +

+∂ b ξ r ; t, x

∂ uv

1 r , ξ r ; t, x ξ

2 r ; t, x .

Lemma 2.3.Let the functions ξ s ; t , x and u t , x exist and be in themean in all their arguments and satisfy, for allq ≥ 2

x

sup E | ξ s ; t, x |q

+ | u t, x |q < ∞

Assume that coefficients of system (2.1) have continuous bounded second-order

derivatives in x andu . Then the second-order derivativesξ 2

r ; t , x = ∂ 2 ξ

∂ x 2 ,


u 2

t , x = ∂ 2 u

∂ x 2 exist in the sense of convergence in the mean, and they satisfy

the system

u 2

t , x = Φ 2

0 ; t , x +t

0

∫ F 2

r ; t , x d r +

+t

0

∫ G 2

r ; t , x d←w 1 r #

ξ 2

s ; t, x =t

s

∫ B 2

r ; t, x d r +

+s

t

∫ E C 2

r ; t , x / Ϝ 0 , t d←w r

and

x

sup E | u 2

t , x |p

+ | ξ 2

s ; t , x |p < ∞

If the second order partial derivatives of the coefficients of the system (2.1) satisfy Höldercondition with exponentγ ∈ 0 ,1 , then

x ≠ y

sup E |u

2 t , x − u

2 t , y

| x − y |γ |

p+

+ |ξ

2 s ; t , x − ξ

2 s ; t , y

| x − y |γ |

p < ∞ #

The coefficients in system (2.6) are obtained from those of (2.1) by means of repeatedformal differentiation with respect to parameterx . In contrast to the corresponding assertion

for the first order derivatives, the a priori existence ofu 2

t , x is not assumed. Theother statements in the lemma are proved similarly to the preceding one and so it will not be

done. To show thatu 2 t , x , ξ 2

s ; t , x exist it is necessary to repeat theproof of Theorem 2.1 with some minor additions.

We pause now to consider the question of relationship between a solution to the system(2.1) and quasilinear parabolic equations.

Lemma 2.4.Let u t , x , ξ s ; t , x be a solution to system (2.1). Then


ξ s ; t , x = ξ s ; r , ξ r ; t , x

#

for any r ∈ s , t with probability 1.Proof.From the system (2.1), we obtain

ξ s ; t, x = ξ r ; t, x +r

s

∫ b l, ξ l ; t, x , v l , ξ l ; t, x d r −

−r

s

∫ E c l, ξ l ; t, x , v l, ξ l ; t, x / Ϝ l , t d←w r

Note, that the random functionξ s ; r , ξ r ; t , x is measurable in all theargumentss , t , x being the composition of measurable mappings, and is consistent with theflow of σ-algebras Ϝ s , t and

ξ s ; r , ξ r ; t , x = L s ; t , ξ r ; t , x ξ , u ,

where the operator L s ; t , x was defined in the proof of Theorem 2.1.Therefore

E | ξ s ; t, x − ξ s ; r , ξ r ; t , x |2 ≤

≤ 2 L2 1 + T

r

s

∫ 1 +x ≠ y

sup E |u l , x − u l , y

x − y|

2 ×

× E | ξ l ; t, x − ξ l ; r , ξ r ; t , x |2d l

Taking into account estimate (2.2) and applying Gronwall’s lemma,we can easilycomplete the proof of Lemma2.4.

Random functionξ s ; t, x is continuous over all its arguments, and so has a

separable modification such that equality ( 2.8) holds with probability 1 over the introducedtime interval. In what follows this modification we will meanconsideringξ s ; t, x . Fromlemma 2.4 ensue that


ϕ ξ 0 ; t, x +s

0

∫ f r, ξ r ; t, x , v r , ξ r ; t, x d r +

+s

0

∫ g r, ξ r ; t, x d←w 1 r = ϕ ξ 0 ; s, ξ s ; t, x +

+s

0

∫ f r, ξ r ; s, ξ s ; t, x , v r , ξ r ; s, ξ s ; t, x d r +

+s

0

∫ g r , ξ r ; s, ξ s ; t, x d←w 1 r = ϕ ξ 0 ; s , z +

+s

0

∫ f r , ξ r ; s, z , v r , ξ r ; s, z d r +

+s

0

∫ g r , ξ r ; s, z d←w 1 r z = ξ s ; t, x

Correctness last equality follows both from the fact that the expression inside the bracketsis continuous and therefore measurable inz and thatξ s ; t, x is independent onσ-algebra Ϝ 0 . s . Hence,

u s, ξ s ; t, x = ϕ ξ 0 ; t, x +

+s

0

∫ f r, ξ r ; t, x , v r, ξ r ; t, x d r +

+s

0

∫ g r, ξ r ; t, x d←w 1 r

Theorem 2.3.Suppose that the hypotheses of Theorem 2.1 hold, the solution to system

(2.1) is differentiable in the mean and satisfy (2.7). Then the function v t , x is a solutionof Cauchy problem for the equation


v t, x = ϕ x +t

0

∫ b s, x, v s, x , ∇ v s, x +

+ 12

C s, x, v̄ s, x C∗ s, x, v̄ s, x ∇ , ∇ v s, x +

+ f s, x, v s, x d s +

−t

0

∫ C 1 s, x, v̄ s, x d w 1 s , ∇ v s, x −t

0

∫ g s, x d w 1 s

where

C s , x , v = ∑ i = 1

2c i

∗ s , x , v c i s , x , v

12 ,

C s, x, v̄ = E C s, x, v s, x / Ϝ s , t = E C s, x, v s, x ,

c 1 s, x, v̄ = E c 1 s, x, v s, x / Ϝ s , t = E c 1 s, x, v s, x

Proof. It follows from Lemma 2.4 that

u t, x = u s, ξ s ; t, x +t

s

∫ F r d r +t

s

∫ G r d w 1 r

where

F r = f r , ξ r ; t, x , v r , ξ r ; t, x

and the functionsG r , C r , B r are defined similarly. Lets = t 0 < t 1 < ... < t n = t be a partition of the interval s , t and letλ = max Δ t i ,

Δ t i = t i + 1 − t i . Then


u s , ξ s ; t, x − u s , x =

=n − 1

i = 0

∑ u s, ξ t i + 1 ; t, x − u s, ξ t i ; t, x

Since in what followst and x will be viewed as fixed parameters, we shall writeξ s instead of ξ s ; t, x . Since u t , x is sufficiently smooth, applying Taylor’stheorem, we have

n − 1

i = 0

∑ u s , ξ t i + 1 − u s , ξ t i =

=n − 1

i = 0

∑ u 1

s , ξ t i + 1 ξ t i + 1 − ξ t i +

+ 12

Tr ξ t i + 1 − ξ t i ∗

u 2

s, ξ t i + 1 ξ t i + 1 − ξ t i +

+1

0

∫ 1 − θ 12

Tr ξ t i + 1 − ξ t i ∗ u

2 s, ξ θ t i + 1 −

− u 2

s, ξ t i + 1 ξ t i + 1 − ξ t i d θ = I 1 + I 2 + I 3 ,

where ξ θ t i + 1 = 1 − θ ξ t i + θ ξ t i + 1 . Using the properties

of conditional expectations, we see that

E | I 3 | ≤x ≠ y

sup E |u

2 t , x − u

2 t , y

| x − y |γ | ×

× ∑ i E | ξ t i + 1 − ξ t i |2 + γ ≤ R t − s λ

γ2

where the constantR does not depend on the partition made. Consider now the quantity

I 1 . We have


I 1 =n − 1

i = 0

∑ u 1

s, ξ t i + 1 B t i + 1 Δ t i +

+ E C t i + 1 / Ϝ i + 1 , n Δ w← t i + o 1 λ ,

where subscript i stands byt i , and

Δ←

w t i = w t i − w t i + 1 ,

o 1 λ =n − 1

i = 0

∑ u 1

s , ξ t i + 1

i + 1

i

∫ B r − B t i + 1 d r +

+

i + 1

i

∫ E C r / Ϝ r t − E C t i + 1 / Ϝ i + 1 , n d←w r .

Let us estimateo 1 λ

E | o 1 λ |2 ≤ 2 λ t − s

x

sup E | u 1

t, x |2

| r − q | < λ

sup E | B r, ξ r , v r, ξ r − B q, ξ q , v q, ξ q |2

+

#

+ 2 E |

n − 1

i = 0

∑ u 1

s, ξ t i + 1

i + 1

i

∫ E C r / Ϝ r t

− E C t i + 1 / Ϝ i + 1 , n d←w r |

2

.To estimate the first term here under the expectation , we first took the expectation with

respect to Ϝ s , t and then took into account that the random variablesξ t i + 1 and


i + 1

i

∫ B r − B t i + 1 d r are Ϝ s , t - measurable for each

i = 0, 1, ...n − 1 and that u s , ∗ does not depend on thisσ- algebra. We pause toestablish the bound for the second term in detail. Leti < k . Taking first the conditionalexpectation with respect toϜ i , n , we obtain

E u 1

s, ξ t i + 1

i + 1

i

∫ E C r / Ϝ r t −

− E C t i + 1 / Ϝ i + 1 , n d←w r

∗u

1 s , ξ t k + 1 ×

×

k + 1

k

∫ E C r / Ϝ r t − E C t k + 1 / Ϝ k + 1 , n d←w r =

= E E

i + 1

i

∫ E C r / Ϝ r t − E C t i + 1 / Ϝ i + 1 , n d←w r

∗×

× u 1

s, z i + 1 ∗

u 1

s , z k + 1 / Ϝ i + 1 , n z i + 1 = ξ t i + 1

zk + 1 = ξ t k + 1

×

×

k + 1

k

∫ E C r / Ϝ r t − E C t k + 1 | Ϝ k + 1 , n d←w r

The expression under the conditional expectation sign does not depend onthe σ-algebraϜ i + 1 , n and therefore the conditional expectation coincides with the expectation for which,

for any i

E u 1

s, z i + 1

i + 1

i

∫ E C r / Ϝ r t −

− E C t i + 1 / Ϝ i + 1 , n d←w r

∗u

1 s , z k + 1 =


= E

i + 1

i

∫ E C r / Ϝ r , t − E C t i + 1 / Ϝ i + 1 , n d←w r

∗×

× E u 1

s , z i + 1 ∗

u 1

s, z k + 1 = 0 .

Taking this into account, we observe that the second term on theright - hand side of (2.8)

does not exceed

2 t − s y

sup E | u 1

t, y |2E

| r − q | < λ

sup E | C r, ξ r , v r, ξ r −

− C q, ξ q , v q, ξ q |2

Since

E | C r , ξ r , v r , ξ r − C q , ξ q , v q , ξ q |2 ≤

≤ 3x , u

sup E | C r ,x, u − C q, x, u |2 + 2K2z

sup E | u r, z − u q, z |2 +

+ K 1 + 2 K 2 sup E |u s, x − u s, y

x − y|

2 E | ξ r − ξ q |

2

where K 1 =t , x , u

sup |∇ x b t , x , u |2, K 2 =

t , x , u

sup |∇ u b t, x, u |2. It is

now not hard to see from this thatP . lim o 1 λ = 0 . Then

I 2 =

i = 0

n − 1

∑ 12

Tr E C∗ t i + 1 / Ϝ i + 1 , n u

2 s, ξ t i + 1 ×

× E C t i + 1 / Ϝ i + 1 , n Δ t i + o 2 λ , t − s ;

where


o2 λ, t − s = 12 ∑

i

Tr

i + 1

i

∫ B r d r +

i + 1

i

∫ EC r / Ϝ r t d←w r ∗ ×

× u 2

s, ξ t i + 1

i + 1

i

∫ B r d r +

i + 1

i

∫ E C r / Ϝ r t d←w r −

− E C∗ t i + 1 / Ϝ i + 1 , n u

2 s , ξ t i + 1 ×

E C t i + 1 / Ϝ i + 1 , n Δ t i =12 ∑ i Tr

i + 1

i

∫ B∗ r d r ×

× u 2

s, ξ t i + 1

i + 1

i

∫ B r d r +

i + 1

i

∫ B∗ r d r u 2 s, ξ t i + 1 ×

×

i + 1

i

∫ E C r / Ϝ r t d←w r +

i + 1

i

∫ E C∗ r / Ϝ r t d

←w r ×

× u 2

s, ξ t i + 1

i + 1

i

∫ B r d r +

i + 1

i

∫ E C∗ r / Ϝ r t d

←w r ×

× u 2

s, ξ t i + 1

i + 1

i

∫ E C r / Ϝ r t d←w r −

− E C∗ t i + 1 / Ϝ i + 1 , n u

2 s , ξ t i + 1 ×

× E C t i + 1 / Ϝ i + 1 , n Δ t i = o 21 λ + o 22 λ + o 23 t − s


We estimate each of the last three quantities. Taking first theconditional expectation with

respect to theσ-algebraϜ s t and using the independence of the random function

u s ,∗ with respect to thisσ-algebra, we find easily that

E | o 21 λ | ≤ λ t − s N 8 , E | o 22 λ | ≤ λ12 t − s N 9

whereN 8 , N 9 are positive constants not depending ont − s , λ . Next

E | o 23 t − s | ≤ 12 z

sup E | u 2

s, z |

i

∑

i + 1

i

∫ E C r / Ϝ r, t d←w r | 2 −

− Tr E C∗ t i + 1 C t i + 1 / Ϝ i + 1 , n Δ t i

It is well known that the expression in the brackets being summed approaches 0 and it isnot hard to see that

t → s

lim t − s − 1

E | o 23 t − s | = 0 .

Thus

I 1 + I 2 = ∑i

u 1

s, x b s, x, v s, x Δ t i +

+ c s, x, v̄ s, x Δ←w t i +

+ 12

Tr c s, x, v̄ s, x u 2 s, x c∗ s, x, v̄ s, x Δ t i + o 3 t − s

where

o 3 t − s =

i

∑ u 1 s, x B t i + 1 − u 1 s, x b s, x, v s, x +

+ 12

Tr E C∗ t i + 1 / Ϝ i + 1 , n u

2 s, ξ t i + 1 ×


× E C t i + 1 / Ϝ i + 1 , n −

− 12

Tr c∗ s, x, v̄ s, x u 2 s , x c s, x, v̄ s, x Δ t i +

+ u 1

s, ξ t i + 1 E C t i + 1 / Ϝ i + 1 , n −

− u 1

s, x c s, x, v̄ s, x Δ←w t i = o 31 + o 32 + o 33 .

We estimate first term ofo 33 which holds the increments of the Wiener process.Similarly to the estimationo 1 , we have

E | o 33 |2 ≤ 1

3E |

i

∑ u 1

s , ξ t i + 1 −

− u 1

s , x E C t i + 1 | Ϝ i + 1 , n Δ←w t i |

2+

+ 2 E | ∑ i u 1

s , ξ t i + 1 E C t i + 1 / Ϝ i + 1 , n −

− c s, x, v̄ s, x Δ←w t i |

2= 2 K

2

i

∑ E | u 1

s, ξ t i + 1 −

− u 1

s, x |2Δ t i + 2

z

sup E | u 1

s , z |2

×

×

i

∑ E | E C t i + 1 / Ϝ i + 1 , n − c s, x, v̄ s, x |2Δ t i ≤

≤ 2 K2 t − s

x ≠ y

sup E |u

1 t, x − u

1 t, y

x − y|

2×

×r < t

sup E | ξ r ; t , x − x |2+ 2 t − s

zsup E | u

1 s, z |

2×


×r < t

sup E | c r , x , v r , ξ r ; t , x − c s , x , v̄ s , x |2

Applying the hypotheses of the theorem and formula (2.2) , we obtain

t → s

lim t − s − 1

E | o 33 | = 0 . Next

E | o32 | ≤ 12

i

∑ Tr E | E C∗ t i + 1 − c∗ s, x, v̄ s, x / Ϝ i + 1 , n ×

× u 2 s, ξ t i + 1 E C t i + 1 / Ϝ i + 1 , n + E | c∗ s, x, v̄ s, x ×

× u 2

s, ξ t i + 1 − u 2

s, x E C∗ t i + 1 / Ϝ i + 1 , n |+

+ E | c∗ s, x, v̄ s, x u

2 s, x E C t i + 1 −

− c s, x, v̄ s, x / Ϝ i + 1 , n | Δ t i

The first and third terms can be estimated in the same way and they do not exceed the quantity

t − s K2

zsup E | u

2 s , z |×

×r < t

sup E | c r , x , v r , ξ r ; t , x − c s , x , v̄ s , x

which was shown above to approach to 0 ast → s . It also not hard to see that thesecond term does not exceed

t − s K2

p ≠ q

sup E |u

2 s, p − u

2 s, q

| p − q |γ |

r < t

sup E | ξ r ; t, x − x |γ

which approach to 0 ast → s . Hence,

t → s

lim t − s − 1

E | o 3 t − s | = 0 .

Now, let 0 , t be a time interval in which smooth solution to the system (2.1) exists.Put 0= s 0 < s 1 < ... < s l = t and λ = max i Δ s i . Applying the


expansions and estimates obtained above to each interval s i , s i + 1 , we see that

u t, x − u 0, x =l − 1

i = 0

∑ u s i + 1 , x − u s i , x =

=l − 1

i = 0

∑ u 1

s i , x b s i , x, v s i , x Δ s i +

+ c s i , x, v̄ s i , x Δ←w s i +

+ 12

Tr c∗ s i , x, v̄ s i , x u 2 s i , x c s i , x, v̄ s i , x Δ s i +

+ F s i Δ s i + G s i Δ w 1 s i + o λ .

whereλ → 0

lim λ− 1

E | o λ | = 0 . Taking the conditional expectation with respect to

theσ-algebraϜ 0 s

1 and then lettingλ → 0 , we complete the proof of the theorem.

Here, we represent other type of Stochastic Partial Differential Equations. These

evolution systems arise in studying of a media in which macro parameter changed with respectto the distributions of the characteristics. The technics which we need to apply very similar tothe one that described above. Consider the system

ξ s ; t, x = x +t

s

∫ b r, ξ r ; t, x , v r, ξ r ; t, x d r −

#

−t

s

∫ E c r, ξ r ; t, x , v r, ξ r ; t, x / Ϝ r , t d←w r

u t, x = E ϕ ξ 0 ; t, x +

+t

0

∫ E f r, ξ r ; t, x , u r, ξ r ; t, x / Ϝ 0 r d r +


+t

0

∫ E g r, ξ r ; t, x , u r, ξ r ; t, x / Ϝ 0 r d w r

Note that Ito integrals involved in the system (2.10) are backward and forward

respectively.Theorem 2.4Let coefficients of the system (2.10) satisfies the conditions ofthe theorem

2.2. Then

v t , x = ϕ x +t

0

∫ b s, x, u s, x , ∇ u s, x +

+ 12

C s, x, ū s, x C∗ s , x , ū s , x ∇, ∇ u s, x +

+ f s, x, u d s +t

0

∫ g s, x, u d w 1 s

Stochastic Shrödinger Equations .

In this section we briefly represent a class SPDE that have not discussed above. This isStochastic Shrödinger Equations. There are two peculiarities thatdiffer Shrödinger Equationsfrom studied at previous sections. They are linearity of the equations and complexity of thecoefficients. The linearity simplifies some standard analytic tools used usually for the proofnonlinear systems but from other hand complex coefficients make additional mathematicaldifficulties. We introduce the complex representation for the Shrödinger equation solution.This representation was first given in [9]. Some applications of this approach to the particularproblems were studied in [16,17,18, 23-26,29-31]. It is clear that having two differentrepresentations for the wave equation is impossible. In this paper we represent the proof ofequivalence of the complex space and Feynman representations. Then choosing nonstandardtype of Lagrangian for the particle in the potential field we obtainthe probabilistic density ofthe particle. This idea is based on using other phase coordinates for the classical particlemovement. The first step on the way to quantizing a system entails rewriting the problem inLagrangian form. The Lagrangian for a system ofN − point particles with massesm j , j = 1 , 2 , ....N moving in a potentialV t, x is


L x,d x

d t, t = 1

2

N

j = 1

∑ m j d x j

d t2 − V x, t #

Here x = x 1

, x 2

, x 3

is a point in Euclidean space. Thus

x j t describes the motion of thejth particle andd x j

d tdetermines its velocity along the

path in the space. We denote a probability amplitudeψ x, t of finding the particle at thelocation x and at timet. With K x, s | y, t we introduce the transition amplitude for theparticle that is emitted atx at the times and is being detected aty at time t. The totalamplitudeψ y, t reads

ψ y, t = ∫ K x, s | y, t ψ x, s dx

#

This fundamental dynamical equation of the quantum theory and it is completely equivalent

to Shrödinger equation. The main concern is to find kernelK. Following the idea of Dirac’sFeynman used expression

K x, s | y, t = ∫x s = x

x t = y

dx∗ exp ih ∫ L x r ,

d x r

d r, r dr

#

where right hand integral may be represented in the form

ε → 0

lim ∫ ... ∫ expi

hS

d x1

A...

d xN − 1

A #

A = 2 π i h

m

12

and actionS =t

s

∫ L x, dxdr

, r dr . The complex-valued functionψ x, t satisfies

the Shrödinger equation

i h∂ ψ

∂ t=

N

j = 1

∑ h2m j

∂2 ψ

∂ x2+ V x, t ψ

#


and |ψ x, t | 2 represents the probability density of presence of the particles at pointx = x 1 , ..., x N at timet. First we suggest to establish a correspondence between the

solution of ( 3.5 ) and diffusion in complex space. Putting

i = − 1 ; z = x + i y ; x, y ⊂ E3

define a random process

ξ j s ; t, z j = z j + i h

m j

12 w j s − w j t = #

= x j + i h

m j

12 w j s − w j t + i y j +

i h

m j

12 w j s − w j t

where w j t are mutually independent 3-dimensional real Wiener processes. Formula( 3.6 ) shows that for givenz j = x j + i y j , t ≥ 0 the processξ j s ; t, z j has

linear manifold as its phase space. This manifold consists of aset of direct lines in each of the

complex planes of variablez j

k = x j

k + i y j

k under the angleπ

4to each

coordinate axis and going through the point x j

k , y j

k . From the form of the function

ξ j ∗ it follows that it is analytic function with respect to complex variable with zprobability 1. By analogy with classical physics , the complex randomprocessesξ j s ; t, z j are characteristics of equation (3.5), along which the macroparameterψ takes its values. This unique macroparameter represent complete information about aquantum system and give us possibility to assert that the quantum systemis really exist in thecertain state. This confidence is equal to the one that the function x s ; t , y determinedby (0.1) as the function in times is able to present the complete description of the classicalmovement in classical mechanics.

Theorem 3.1Assume that nonrandom vector and scalar functionsΨ 0 z , V z , t defined

for t, z ⊂ 0, + ∞ × Z 3 N = Z +3 N are analytic with respect to thez and

continuous int . Then the function

Ψ z , t =

#

= E Ψ 0 ξ 0 ; t, z exp ih ∫

o

tV ξ s ; t, z , s ds

is a classical solution of the Cauchy problem


∂ Ψ z, t

∂ t=

N

j = 1

∑ i h

2m j

∇j2 Ψ z, t − i

hV z, t Ψ z, t

#

Ψ 0 , z = Ψ 0 z

Note that analyticity of the functionΨ z , t follows from the identity

∂Ψ z ,−z

∂_

Z= 0 , where

−z = x − i y. Put for simplicity N = 1 and V = 0

P t, x, y =

= E Re Ψ x + h2 m

12 w s − w t , y + h

2 m

12 w s − w t , t ;

Q t, x, y =

= E Im Ψ x + h2 m

12 w s − w t , y + h

2 m

12 w s − w t , t .

Recall if Ψ t , z = P t , x , y + i Q x , y be the analytic function of thevariable z then Cauchy Riemann equations hold

∂ P

∂ x=

∂ Q

∂ y,

∂ P

∂ y= −

∂ Q

∂ x

From these equations we can justify that

h

4 m

∂∂ x

+∂∂ y

2

P t, x, y = −h

2 m

∂ 2Q t, x, y

∂ x2

h

4 m

∂∂ x

+∂∂ y

2

Q t, x, y =h

2 m

∂ 2P t, x, y

∂ x2

Hence,

∂∂ t

P + i Q =i h

2 m

∂ 2

∂ x2 P + i Q


Putting z = x + i 0 in (3.8 ) we can verify thatψ x, t = Ψ x + i 0, 0 is asolution of Shrödinger equation andψ x, 0 = Ψ x + i 0 . Thus Shrödinger equation

may be interpreted as a trace of complex Kolmogorov equation on the real part of theZ3 N

.It should be noted that the condition

∫ | ψ x , t |2

dx = 1

is natural for the functionψ x , t . Setting t = 0 we show possibility of theanalytical extension of the initial wave function.

Theorem Polya , Plancherel

In order that the complex functionφ z be an interger of an exponential type and∫ | φ x + i 0 |

2dx < ∞ , it is necessary and sufficient that following representation

take place

φ z = 2 π − n

2

E n

∫ Φ q exp − i q , z dq

whereΦ belong toL 2 En and has compact support . Let introduce backward real -

valued vector Wiener process

η s; t, x = x + w s − w t

Then the processξ ∗ can be expressed as

ξ s ; t, x = x + i h

m

12 η s ; t , x − x

Then using ’backward ’ Markov property for the processη and formula ( 3.7 ) we canwrite representation

ψ x , t =ε → 0

lim exp ih

V x , t × #

× ∫ expi h

V x + i h

m

12 xN−1 − x , tN−1 p t, x, tN−1, xN−1 d xN−1 ×


× ∫ exp i h

V x + i h

m

12 x 1 − x , tN−1 p t2 , x2 , t1 , x1 d x1 ×

∫ ψ 0 x0 + i h

m

12 x0 − x p t1 , x1 , t0 , x0 d x0

where = tk+1 − tk , k = 0 , 1 , 2 , ....,N − 1 , s = t0 < t1 < t2 < .... < tN = t

a partition of interval s , t and

p t , x , s , y = 2 π t − s − 3 N

2exp −

x − y 2

2 t − s

Note that obtained formula (3. 9 ) differs from Feynman’s representation (3. 2), ( 3.3 ), ( 3.4 ).

Theorem 3.2Representations ( 3.2-4) and ( 3.9 ) are equivalent.

Proof. For simplicity put V = 0, and assume thatN = 1 and dimension of the

coordinate space is also equal to 1. In this case from (3.9) we have

ψ t , x = E φ ξ s ; t , x =

#

2 π t − s − 1

2 ∫ φ x + i hm

12λ exp −

λ2

2 t − s dλ

Denote Fourier transformation

Φ y = 12 π

+ ∞

− ∞

∫ e− i x y

φ x dx

φ x =

+ ∞

− ∞

∫ ei x y

Φ y dy


and suppose thatΦ y has compact support. Substituting Fourier transformation into (3.10 ) and change the order of integrals we have

ψ t , x = 2 π t − s − 1

2 ∫ exp −λ

2

2 t − s d λ ∫ Φ y

exp x + i hm

12λ y dy =

= 2 π t − s − 1

2 ∫ Φ y dy ∫ exp −λ

2

2 t − s +

+ i x − i hm

12 λ y dλ =

= 2 π t − s − 1

2 ∫ Φ y exp i x y dy ∫ exp −λ

2

2 t − s +

+ −i hm

12λ y dλ =

= ∫ Φ y exp i −h t − s y

2

2 m+ x y d y 2 π t − s

− 12×

× ∫ exp − 12

λ

t − s+ y

i h t − s

m

2

dλ =

= ∫ Φ y exp i x y −h t − s y

2

2 m dy

Taking into account equality

exp −h t − s y 2

2 mΦ y = 2π − 1 ∫ φ λ exp − i λ y +

+ 2 m − 1

h t − s y2 d λ

we note that


∫ Φ y exp i x y −h t − s y

2

2 m dy =

= 2π − 1 ∫ φ λ dλ ∫ exp i y x − λ − 2 m − 1

h t − s y2 d y

Calculating the inner integral we obtained

2π − 1 ∫ exp i y x − λ − 2 m − 1

h t − s y2 dy =

= 2 π − 1 ∫ exp − 1

2

i h t − s m

12 y −

i m

h t − s

12 x − λ

2

×

× expi m x − λ

2

2 h t − s d y = 1

2 π

2 π m

i h t − s

12

expi m x − λ

2

2 h t − s

Hence,

ψ x , t = m2 π h t − s

12 ∫ exp

i m x − y 2

2 h t − s φ y dy

Now return to equation (3.1). Let potential function be generated by an external forces

”white noise type:

V x , t , ω = f t , x + g t °β t , x =

=N

j = 1

∑ f j t x j +

j = 1

N

∑d

k = 1

∑ g j k t x j

°β k t = F x, t + G x, t

°β t

where β t is d- dimensional Wiener process , independent ofw t . Our goal towrite down in correct form the Shrödinger equation using probabilisticrepresentation of thesolution. Following what has been stated above , we introduce the functional

Ψ z , t = E Ψ 0 ξ 0 ; t, z ×

#


× exp ih

t

0

∫ F ξ s ; t, z , s ds +t

0

∫ G ξ s ; t, z , s d←β s / Ϝ

tβ

where Ϝ t

β= σ β s − β l ; s , l ∈ 0 , t . Let shortly discuss the

representation of the wave function (3.11). In view of the fact thatthis formula can be used forthe probabilistic representation, we conclude that randomness caused by the potential mayremain in the Shrödinger equation if and only if the potential and quantum particle trajectoriesare mutually independent. In this case, a solution of Shrödinger equation can be represented asa conditional expectation with respect to theσ −algebra generated by the random potential.Now we derive the Shrödinger Equations for the wave function (3.11) and then verify the

normalization condition.Theorem 3.3

Suppose that nonrandom functionΨ 0 z , F z , t , G z , t are continuouswith respect tot and analytic with respect toz. Then the functionψ x , t = Ψ x + i 0 , t is a classical solution of the Cauchy problem for theShrödinger equation

∂ ψ

∂ t=

i h

2 m∇ 2 ψ x, t + h

− 1 i F x, t − h

− 1G x, t ψ x , t −

− i h− 1

G t , x ψ x , t °β t ,

#

ψ x , 0 = Ψ 0 x + i 0

The classical solution of the Cauchy problem (3. 12) is a random functionψ x , t twice continuously differentiable with respect tox in the sense of mean, which ismeasurable with respect toϜ t

βand satisfies the equality

ψ x , t − ψ x , 0 =

#

=t

0

∫ i h

2 m∇ 2 ψ x, s + h

− 1 i F x, s − h

− 1G x, s ψ x, s d s −


− i h− 1

t

0

∫ G x , s ψ x , s d β t

for all t with probability 1. In equality ( 3.13 ), the Ito integral is the forward course oftime.

Proof. When deriving equation ( 3.13 ) we use the same method as for the real case sojust sketch the basic ideas of the proof. Lets = t 1 < t 2 < ... < t N = t be anarbitrary patition of the interval s , t and λ =

jmax Δ t j . The processes

ξ s ; t, z and β t are mutually independent, the flowϜ t

βis continuous int , and

ξ s ; t, z is a diffusion process. Taking these into account, we obtain

Ψ z , t k + 1 − Ψ z , t k = E Ψ ξ t k ; t k + 1, z , t k + 1 −

− Ψ z, t k exp ih

N

j = 1

∑ f j t k z j Δt k −

j = 1

N

∑d

l = 1

∑ g j l t k z j Δβ l t k +

+ Ψ z , t k exp ih

N

j = 1

∑ f j t k z j Δ t k −

−

j = 1

N

∑d

l = 1

∑ g j l t k z j Δ β l t k / Ϝ k+1

β − 1 + o λ

whereϜ k

β= Ϝ t k

β. By using the fact thatσ- algebras Ϝ k , k+1

βand

Ϝ k , k+1

ξ ∨ Ϝ 0 , k

βare conditionally independent , we have

Ψ z, t k + 1 − Ψ z, t k = E ξ t k ; t k + 1, z − z ∇ z

2Ψ z, t k +

+ ξ t k ; t k + 1, z − z ∇ z

2Ψ z, t k ξ t k ; t k + 1, z − z

∗+


+ Ψ z , t k ih

N

j = 1

∑ f j t k z j Δ t k −

− 12 h

2

j = 1

N

∑d

l = 1

∑ g j l t k z j Δ β l t k Δ t k −

− ih

j = 1

N

∑d

l = 1

∑ g j l t k z j Δ β l t k + o λ / Ϝ k

β.

If we now sum over allk = 1 , 2 , ......N − 1 and pass to the limit in probability as

λ → 0 , we get

Ψ t, z − Ψ s, z =t

s

∫ i h2m

∇ z

2Ψ z, t k + i

hΨ z, r F z, s −

#

− 12 h

2 G2 z, r Ψ z , r d r −

ih

t

s

∫ G z, r Ψ z, s d β t

Here, we setz = x + i 0 and s = 0 and take into account that ∇ z

2

Ψ z, t k = ∇ x

2Ψ z, t k . This yields equality (3.13).

Theorem 3.4Under the conditions of the theorem 3.3. the normalizing condition

∫ | ψ x , t |2

d x = ∫ | ψ x , 0 |2

d x

holds for anyt ≥ 0 .Proof.For convenience, rewrite ( 3.14 ) in the differential form

∂ sΨ z, s = i h2m

∇ z

2Ψ z, t k + i

hΨ z, s F z, s −


− 12 h

2 G2 z, s Ψ z, s d s −

ih

G z, s Ψ z, s d β s

where ∂ s is the differential with respect tos . Hence, the complex conjugate function

Ψ z , s satisfies equality

∂ sΨ z , s = −i h2m

∇ z

2Ψ z , s − i

hΨ z , s F

−z, s −

− 12 h

2 G2

−z, s Ψ z , s d s + i

hG

−z, s Ψ z , s d β s

Treating z as a fixed parameter , we use the stochastic formula of integrationbyparts. Then

∂ s | Ψ z , s |2= ∂ sΨ z , s Ψ z, s + Ψ z , s ∂ sΨ z, s +

+ ∂ sΨ z , s ,Ψ z, s > = i h2m

Ψ z , s ∇ z

2Ψ z, t k + | Ψ z, s |2 ×

× F z, s − 12 h

2 G2 z, s | Ψ z, s |

2 d s − i

hG z, s ×

× | Ψ z, s |2d β s + −

i h2m

Ψ z , s ∂ 2

Ψ z , s

∂ z2 −

− ih

| Ψ z, s |2

F −z, s − 1

2 h2 G

2

−z, s | Ψ z, s |

2 d s +

+ ih

G _z, s | Ψ z, s |

2d β s + 1

h2 G

2 z, s | Ψ z, s |

2d s

Setting z = x + i 0 and ψ x , t = Ψ x + i 0 , t we cancel the similarterms. By integrating the equality obtained with respect to the variablex over ball S R ofradiusR , we get


S R

∫ ∂∂ t

| ψ x , t |2

d x =

=i h

2 mS R

∫ div −ψ x , t ∇ ψ x , t − ∇

−ψ x , t ψ x , t d x =

=i h

2 m∂ S R

∫ div −ψ x, t ∇ ψ x, t − ∇

−ψ x, t ψ x, t n d S R

where d S R is an element of the surface of the ballS R and k n is the projection ofthe vectork onto the outer normal to∂ S R . Note the integrand is bounded and continuous inthe variablest andx. The expression on the right hand side has standard form obtained whenchecking the normalization condition in the case of a deterministic potential. Assuming thatconditions guaranteeing the convergence of the surface integral to 0 as R → ∞ are satisfiedwe pass to the limit and complete the proof.

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