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Chaos expansions: applications to ageneralized eigenvalue problem for theMalliavin derivativeTijana Levajković a , Stevan Pilipović b & Dora Seleši b

a Department of Mathematics and Informatics, Faculty of Trafficand Transport Engineering , University of Belgrade , Vojvode Stepe305, 11000, Belgrade, Serbiab Department of Mathematics and Informatics, Faculty ofScience , University of Novi Sad , Trg Dositeja Obradovića 4,21000, Novi Sad, SerbiaPublished online: 06 Aug 2010.

To cite this article: Tijana Levajković , Stevan Pilipović & Dora Seleši (2011) Chaos expansions:applications to a generalized eigenvalue problem for the Malliavin derivative, Integral Transformsand Special Functions, 22:2, 97-105, DOI: 10.1080/10652469.2010.499734

To link to this article: http://dx.doi.org/10.1080/10652469.2010.499734

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Integral Transforms and Special FunctionsVol. 22, No. 2, February 2011, 97–105

Chaos expansions: applications to a generalized eigenvalueproblem for the Malliavin derivative

Tijana Levajkovica, Stevan Pilipovicb* and Dora Selešib

aDepartment of Mathematics and Informatics, Faculty of Traffic and Transport Engineering, University ofBelgrade, Vojvode Stepe 305, 11000 Belgrade, Serbia; bDepartment of Mathematics and Informatics,

Faculty of Science, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

(Received 27 April 2010; final version received 1 July 2010 )

We study the chaos expansion transform, in short, chaos expansions, in a class of white noise spaceswith series expansions by means of Hermite polynomials and functions, with certain weight sequences.Since Hermite polynomials are eigenfunctions for the Ornstein–Uhlenbeck operator, we apply the chaosexpansion transform in solving of a class of equations. Moreover, we solve a generalized eigenvalueproblem for the Malliavin derivative by means of chaos expansions.

Keywords: Hermit polynomials and functions; series expansions of distributions; generalized randomprocess; chaos expansion; Malliavin derivative; Ornstein–Uhlenbeck operator

AMS Subject Classification: 33E20; 44A05; 46F12; 60H40; 60H15

1. Introduction

The chaos expansion of stochastic processes provides a series decomposition of square integrableprocesses in a Hilbert space with an orthogonal basis built upon a class of special functions,Hermite polynomials and functions, in the framework of white noise analysis. In order to buildspaces of stochastic test and generalized functions, one has to use series decompositions viaorthogonal functions as a basis, with certain weight sequences. These ideas are very similar tothose used by Zemanian in [17, Chapter 9] to build deterministic space of generalized functions(see [1] for the recent applications). White noise analysis, introduced by Hida (cf. [2]) and furtherdeveloped by many authors (see, e.g. [3,5,11] and references therein), has had found applicationsin solving stochastic differential equations [6,8–10,14].

This paper deals with the definitions and applications of the chaos expansion transform, in short,chaos expansion, to a generalized eigenvalue problem for the Malliavin derivative for which theseries with Hermite functions and polynomials are the crucial objects so that the solution hasa simple form within this approach. The Malliavin derivative D appears as the adjoint operatorof the Skorokhod integral δ which is an extension of the stochastic Itô integral of anticipating

*Corresponding author. Email: pilipovic@dmi.uns.ac.rs

ISSN 1065-2469 print/ISSN 1476-8291 online© 2011 Taylor & FrancisDOI: 10.1080/10652469.2010.499734http://www.informaworld.com

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98 T. Levajkovic et al.

processes to the class of non-anticipating processes [13]. In this paper, we also consider thecomposition of these two operators, R = δD, called the Ornstein–Uhlenbeck operator, which isa linear, unbounded and self-adjoint operator. We refer to [3,11–13] for basic results related tothe Malliavin derivative and its applications.

Note that we follow Zemanian [17, Chapter 9], who has studied the transform f → (ak)k ∈ CN0

so that f = ∑k akψk, (ak = (f, ψk) with appropriate dual pairing) and, in our case of chaos

expansions, call in the sequel the chaos expansion transform just chaos expansion.The method of Wiener–Itô chaos expansions, used in equations considered in this paper, is a

general and a useful tool, also known as the propagator method. With this method we reducea problem to an infinite system of deterministic equations. Summing up all coefficients of theexpansion and proving convergence in an appropriate weight space, one obtains the solution ofthe initial equation. Another type of equations investigated by the same method can be found inseveral papers [4–8,14].

The paper is organized in the following manner: in Section 2, we provide some basic informationon chaos expansions and the Malliavin calculus. Section 3 is devoted to the Ornstein–Uhlenbeckoperator in which eigenfunctions are the products of Hermite polynomials (with appropriatearguments) and the corresponding eigenvalues are |α|, α ∈ I. In particular, we study equationP(R)u = f.

In Section 4, we apply our chaos expansion transform in order to solve a generalized eigenvalueproblem with the Malliavin derivative in a general form.

Du = C ⊗ u, Eu = u0, for C ∈ S ′(R), u0 ∈ X, E is the expectation.

2. Notions and notations

Let the basic probability space (�, F, P ) be (S ′(Rn), B, μ), where S ′(Rn) denotes the spaceof tempered distributions, B the sigma-algebra generated by the weak topology on � and μ

denotes the white noise measure given by the Bochner–Minlos theorem. The Bochner–Minlostheorem states the existence of a Gaussian probability measure given by the integral transform

of the characteristic function C(φ) = ∫S ′(Rn)

ei〈ω,φ〉 dμ(ω) = e−(1/2)‖φ‖2L2(Rn) , φ ∈ S(Rn), where

〈ω, φ〉 denotes the usual dual paring between a tempered distribution ω and a rapidly decreasingfunction φ.

Let {ξk, k ∈ N} be the family of Hermite functions and {hk, k ∈ N0} the family of Her-mite polynomials. It is well known that the space of rapidly decreasing functions S(Rn) =⋂

l∈N0Sl(R

n), where Sl(Rn) = {ϕ = ∑∞

k=1 akξk : ‖ϕ‖2l = ∑∞

k=1 a2k (2k)l < ∞}, l ∈ N0, and the

space of tempered distributions S ′(Rn) = ⋃l∈N0

S−l(Rn), where S−l(R

n) = {f = ∑∞k=1 bkξk :

‖f ‖2−l = ∑∞

k=1 b2k(2k)−l < ∞}, l ∈ N0.

Let (L)2 = L2(S ′(Rn), B, μ) and Hα(ω) = ∏∞k=1 hαk

(〈ω, ξk〉), α ∈ I be the Fourier-Hermiteorthogonal basis of (L)2, where I denotes the set of sequences of non-negative integers that haveonly finitely many non-zero components α = (α1, α2, . . . , αm, 0, 0 . . .). In particular, for the kthunit vector ε(k) = (0, . . . , 0, 1, 0, . . .), the sequence of zeros with the number 1 as the kth compo-nent, Hε(k) (ω) = 〈ω, ξk〉, k ∈ N. The length of a multi-index α ∈ I is defined as |α| = ∑∞

k=1 αk .Let a = (ak)k∈N, ak ≥ 1, aα = ∏∞

k=1 aαk

k , aα/α! = ∏∞k=1 a

αk

k /αk! and (2Na)α = ∏∞k=1(2kak)

αk .Note that

∑α∈I a−pα < ∞ if p > 1.

We introduce modified Kondratiev spaces: the space of the Kondratiev stochastic test functionsmodified by the sequence a, denoted by (Sa)1 = ⋂

p∈N0(Sa)1,p, is the projective limit of spaces

(Sa)1,p ={

f =∑α∈I

bαHα ∈ (L)2 : ‖f ‖21,p,a =

∑α∈I

b2α(2Na)pα < ∞

}, p ∈ N0.

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Integral Transforms and Special Functions 99

The space of Kondratiev stochastic generalized functions modified by the sequence a, denotedby (Sa)−1 = ⋃

p∈N0(Sa)−1,−p, is the inductive limit of the spaces

(Sa)−1,−p ={

F =∑α∈I

cαHα : ‖F‖2−1,−p,a =

∑α∈I

c2α(2Na)−pα < ∞

}, p ∈ N0.

For ak = 1, k ∈ N, these spaces reduce to the spaces of Kondratiev stochastic test functions (S)1

and the Kondratiev stochastic generalized functions (S)−1, respectively. We have

(Sa)1 ⊆ (L)2 ⊆ (Sa)−1.

Let X be a Banach space and X′ its dual. The operator norm of a linear bounded mappingA : X → X′ will be denoted by ‖A‖op.

Recall that (S)−1 is nuclear and thus (X ⊗ (S)1)′ ∼= X′ ⊗ (S)−1.

Theorem 2.1 [14] Let X be a Banach space endowed with ‖ · ‖X. Generalized stochasticprocesses as elements of X ⊗ (S)−1 have a chaos expansion of the form

u =∑α∈I

fα ⊗ Hα, fα ∈ X, α ∈ I, (1)

and there exists p ∈ N0 such that

‖u‖2X⊗(S)−1,−p

=∑α∈I

‖fα‖2X(2N)−pα < ∞.

In a similar manner, one can consider processes as elements of X′ ⊗ (S)−1. Note that X′ ⊗ (S)−1

is isomorphic to the space of linear bounded mappings X → (S)−1. With the same notation as in(1), we will denote by Eu = f(0,0,0,...) the generalized expectation of the process u.

Remark 1 Generalized random processes as elements of X ⊗ (Sa)−1 have a chaos expansion ofthe form (1) and there exists p ∈ N0 such that

‖u‖2X⊗(S)−1,−p,a

=∑α∈I

‖fα‖2X(2Na)−pα < ∞.

Example 2.2 Brownian motion is an element of (L)2 and it is defined by the chaos expan-sion Bt(ω) = ∑∞

k=1

∫ t

0 ξk(s) dsHε(k) (ω). The Itô integral of Hermite functions is defined byI (ξk) = ∫ +∞

−∞ ξk(t) dBt(ω) = 〈ω, ξk〉 = Hε(k) (ω). Singular white noise Wt(·) is defined by thechaos expansion Wt(ω) = ∑∞

k=1 ξk(t)Hε(k) (ω), and it is an element of the space (S)−1, for allt . It is integrable and the relation (d/dt)Bt = Wt holds [3] in the (S)−1 sense. The stochasticexponential εh, h ∈ L2(R), defined by εh = exp I (h) = exp(I (h) − (1/2)‖h‖2

L2(R)), belongs to

the space (S)1,p as long as ‖h‖L2(R) is sufficiently small. The chaos expansion of the stochasticexponential is given by εh = ∑

α∈I hαHα(ω), where h = ∑∞k=1 hkξk . We refer to [3,5] for more

details.

2.1. Schwartz space-valued generalized random processes

We give in [15,16] a general setting of S ′-valued generalized random process: S ′(Rn)-valuedgeneralized random processes are elements of X ⊗ (S)−1, where X = X ⊗ S ′(Rn), and are given

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by chaos expansions of the form

f =∑α∈I

∑k∈N

aα,k ⊗ ξk ⊗ Hα =∑α∈I

bα ⊗ Hα =∑k∈N

ck ⊗ ξk,

where bα = ∑k∈N

aα,k ⊗ ξk ∈ X ⊗ S ′(Rn), ck = ∑α∈I aα,k ⊗ Hα ∈ X ⊗ (S)−1 and aα,k ∈ X.

Thus, for some p, l ∈ N0,

‖f ‖2X⊗S−l (Rn)⊗(S)−1−p

=∑α∈I

∑k∈N

‖aα,k‖2X(2k)−l(2N)−pα < ∞.

2.2. The Malliavin derivative within chaos expansion

We give now the definitions of the Malliavin derivative and the Skorokhod integral, which areextension of the definitions of these operators to a space of generalized stochastic processes[6,11–13].

Definition 2.3 Let u ∈ X ⊗ (S)−1 be of the form (1). If there exists p ∈ N0 such that∑α∈I

|α|2‖fα‖2X(2N)−pα < ∞, (2)

then the Malliavin derivative of u is defined by

Du =∑α∈I

∑k∈N

αkfα ⊗ ξk ⊗ Hα−ε(k) .

Operator D is also called the stochastic gradient of a generalized stochastic process u. Theset of processes u such that (2) is satisfied is the domain of the Malliavin derivative, which willbe denoted by Dom(D). The Malliavin derivative D is a linear and continuous mapping fromDom(D) ⊆ X ⊗ (S)−1,−p to X ⊗ S−l(R

n) ⊗ (S)−1,−p, for some p ∈ N0 and l > p + 1, l ∈ N:

‖Du‖2X⊗S−l (Rn)⊗(S)−1,−p

=∑α∈I

∥∥∥∥∥∞∑

k=1

αkfα ⊗ ξk

∥∥∥∥∥2

X⊗S−l

(2N)−p(α−ε(k))

≤∑α∈I

(∑k∈N

α2k ·

∑k∈N

‖ξk‖2−l(2k)p

)‖fα‖2

X(2N)−pα

≤∑α∈I

|α|2(∑

k∈N

(2k)−l+p

)‖fα‖2

X(2N)−pα

≤ C∑α∈I

|α|2‖fα‖2X(2N)−pα < ∞,

where∑

k∈N(2k)−l+p = C for l > p + 1.

Definition 2.4 Let F = ∑α∈I fα ⊗ vα ⊗ Hα ∈ X ⊗ S−p(Rn) ⊗ (S)−1,−p, p ∈ N0, be a gen-

eralized S−p(Rn)-valued stochastic process and let vα ∈ S−p(Rn) be given by the expansionvα = ∑

k∈Nvα,kξk , vα,k ∈ R. Then, the process F is integrable in the Skorokhod sense and the

chaos expansion of its stochastic integral is given by

δ(F ) =∑α∈I

∑k∈N

vα,kfα ⊗ Hα+ε(k) .

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Integral Transforms and Special Functions 101

The Skorokhod integral δ is a linear and continuous mapping δ : X ⊗ S−p(Rn) ⊗ (S)−1,−p →X ⊗ (S)−1,−p. Clearly,

‖δ(F )‖2X⊗(S)−1,−p

=∑α∈I

∥∥∥∥∥∑k∈N

vα,kfα

∥∥∥∥∥2

X

(2N)−p(α+ε(k))

≤∑α∈I

(∑k∈N

v2α,k(2k)−p

)‖fα‖2

X(2N)−pα

=∑α∈I

‖vα‖2−p‖fα‖2

X(2N)−pα < ∞,

because F ∈ X ⊗ S−p(Rn) ⊗ (S)−1,−p, p ∈ N0. We conclude that the image of the Malliavinderivative is included in the domain of the Skorokhod integral. The Ornstein–Uhlenbeck oper-ator R = δD is a linear and continuous mapping from Dom(R) ⊂ X ⊗ (S)−1 into the spaceX ⊗ (S)−1. Note that domains of D and R coincide, i.e. Dom(R) = Dom(D). The Hermite poly-nomials are eigenfunctions of R and the corresponding eigenvalues are |α|, α ∈ I, i.e. RHα =|α|Hα . Moreover, if we apply the previous identity k times successively, we get RkHα = |α|kHα,

k ∈ N.Let a generalized stochastic process u ∈ Dom(D) be given by the chaos expansion u =∑α∈I uα ⊗ Hα, uα ∈ X. Then

Ru =∑α∈I

|α|uα ⊗ Hα.

Clearly, if u ∈ Dom(D) ⊂ X ⊗ (S)−1,−p, then R(u) ∈ X ⊗ (S)−1,−p. This follows from

‖Ru‖2X⊗(S)−1,−p

=∑α∈I

|α|2‖uα‖2X(2N)−pα = ‖u‖2

Dom(D) < ∞.

3. Applications to equations with Ornstein–Uhlenbeck operator

In this section, we will use a simple form of the chaos expansion transform related to the Ornstein–Uhlenbeck operator R and solve a class of stochastic equation in the same spirit as in [17].

Let P(t) = pmtm + pm−1tm−1 + · · · + p1t + p0, t ∈ R, be a polynomial of degree m with

real coefficients. Then, P(R) = pmRm + pm−1Rm−1 + · · · + p1R + p0Id.

Theorem 3.1

(i) Let P be a polynomial such that P(k) �= 0, k ∈ N0. Then, the equation

P(R)u = g, where g ∈ X ⊗ (S)−1,−p for some p > 0, (3)

has a unique solution in X ⊗ (S)−1 given by

u =∑α∈I

gα

P (|α|) ⊗ Hα. (4)

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102 T. Levajkovic et al.

(ii) Let P be a polynomial such that P(k) = 0 for k ∈ M, where M is a finite subset of N0, andlet ci ∈ X, i ∈ M . The equation

P(R)u = g, g ∈ X ⊗ (S)−1, uα = ci, |α| = i, i ∈ M,

has a unique solution in X ⊗ (S)−1, given by

u =∑

α∈I,|α|/∈M

gα

P (|α|) ⊗ Hα +∑

|α|=i∈M

ci ⊗ Hα. (5)

Proof Note that the Hermite polynomials Hα are eigenfunctions also for operator P(R), i.e.P(R)Hα = P(|α|)Hα, α ∈ I holds. Assume that u ∈ X ⊗ (S)−1 is a generalized stochasticprocess of the form (1). Then

P(R)u =∑α∈I

uα ⊗ P(R)Hα =∑α∈I

P(|α|)uα ⊗ Hα. (6)

Thus, P(R) maps Dom(D) ⊂ X ⊗ (S)−1,−p → X ⊗ (S)−1,−p−r for r > 1 + 2m, where r

depends on the growth of P(|α|). Note that for all α ∈ I, |α| �= 0, |P(|α|)| ≤ (2N)mα|α|. Then

‖P(R)u‖2X⊗(S)−1,−p−r

=∑α∈I

‖P(|α|)uα‖2X(2N)−(p+r)α

= |P(0)|2‖u(0,0,0,...)‖2X +

∑α∈I,|α|>0

|P(|α|)|2‖uα‖2X(2N)−(p+r)α

≤ |P(0)|2‖u(0,0,0,...)‖2X +

∑α∈I,|α|>0

|α|2(2N)2mα‖uα‖2X(2N)−(p+r)α

≤ D∑

α∈I,|α|>0

|α|2‖uα‖2X(2N)−pα < ∞,

where D = |P(0)|2 + ∑α∈I,|α|>0(2N)−(r−2m)α , for r > 2m + 1.We can also conclude that P(R)

is a continuous and bounded operator.Let g = ∑

α∈I gα ⊗ Hα , where gα ∈ X, α ∈ I. Then by (6):∑

α∈I P(|α|)uα ⊗ Hα =∑α∈I gα ⊗ Hα. Due to the uniqueness of the Wiener–Itô chaos expansion, the last equation

transforms to the system of deterministic equations P(|α|)uα = gα , for all α ∈ I.Now we prove (i). Since P(|α|) �= 0 for all α ∈ I, it follows that uα = gα/P (|α|) and

Equation (3) has a unique formal solution of the form (4).It remains to prove convergence of the solution in X ⊗ (S)−1,−p, for some p > 0. Note that

there exists C > 0 such that |P(|α|)| ≥ C for all α ∈ I. Thus,

‖u‖2X⊗(S)−1,−p

=∑α∈I

∥∥∥∥ gα

P (|α|)∥∥∥∥2

X

(2N)−pα ≤ 1

C2

∑α∈I

‖gα‖2X(2N)−pα < ∞,

because g ∈ X ⊗ (S)−1,−p. Thus, Equation (3) has a unique solution u ∈ X ⊗ (S)−1,−p.The proof of assertion (ii) simply follows by the previous analysis. The coefficients of the

solution u are given by

uα =⎧⎨⎩

gα

P (|α|) , |α| /∈ M,

ci, |α| = i ∈ M,

and the solution has the form (5) if and only if gα = 0, for |α| ∈ M .Note, if there exists at least one α ∈ M such that gα �= 0, then Equation (3) has no solution. �

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Integral Transforms and Special Functions 103

4. Generalized eigenvalue problem

Consider {Du = C ⊗ u, C ∈ S ′(R),

Eu = u0, u0 ∈ X.(7)

Theorem 4.1 Let C = ∑∞k=1 ckξk ∈ S ′(R). If ck ≥ 1/2k, for all k ∈ N, then the eigenvalue

problem (7) has a unique solution in X ⊗ (Sa)−1, given by

u = u0 ⊗∑

α=(α1,α2,...)∈I

( ∞∏k=1

cαk

k

αk!

)Hα = u0 ⊗

∑α∈I

cα

α!Hα. (8)

Proof Using the chaos expansion method, Equation (7) can be written in the form

∑α∈I

(∑k∈N

(αk + 1)uα+ε(k) ⊗ ξk

)⊗ Hα =

∑α∈I

(∑k∈N

ckuα ⊗ ξk

)⊗ Hα.

Thus, we transform (7) into the system of deterministic equations

(αk + 1)uα+ε(k) = uαck, α ∈ I, k ∈ N. (9)

The solution is obtained by induction with respect to the length of multi-indices α. From Eu = u0

follows u(0,0,0,...) = u0.Let |α| = 0, i.e. α = (0, 0, 0, . . .), Equation (9) reduce to⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

u(1,0,0,0,...) = u0c1

u(0,1,0,0,...) = u0c2

u(0,0,1,0,0,...) = u0c3...

uε(k) = u0ck,

, k ∈ N. (10)

and we receive the coefficients uα for α of length one.Next, for |α| = 1, we have α = ε(i), i = 1, 2, . . .

If α = (1, 0, 0, 0, . . .) then from (9) and (10) we obtain⎧⎪⎪⎪⎨⎪⎪⎪⎩u(2,0,0,0,... = 1

2u(1,0,0,0,...)c1 = 12!c

21u0

u(1,1,0,0,...) = u(1,0,0,0,...)c2 = c1c2u0

u(1,0,1,0,0,...) = u(1,0,0,0,...)c3 = c1c3u0...

(11)

If α = (0, 1, 0, 0, . . .) then from (9) and (10) we have⎧⎪⎪⎪⎨⎪⎪⎪⎩u(1,1,0,0,...) = u(0,1,0,0,...)c1 = c1c2u0

u(0,2,0,0,...) = 12u(0,1,0,0,...)c2 = c2

2u0

u(0,1,1,0,...) = u(0,1,0,0,...)c3 = c2c3u0...

(12)

Continuing with α = ε(k), k ≥ 3, we obtain all uα of length two.

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104 T. Levajkovic et al.

For |α| = 2 from system of equations (9) and the results obtained in the previous steps (11),(12), etc., we obtain uα , for |α| = 3.

We start with α = (1, 1, 0, 0, . . .) and obtain the family⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

u(2,1,0,0,...) = 12u(1,1,0,0,...)c1 = 1

2c21c2u0

u(1,2,0,0,...) = 12u(1,1,0,0,...)c2 = 1

2c1c22u0

u(1,1,1,0,...) = u(1,1,0,0,...)c3 = c1c2c3u0...

then continue with α = (2, 0, 0, . . .) and receive⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

u(3,0,0,0,...) = 13u(2,0,0,0,...)c1 = 1

3!c31u0

u(2,1,0,0,...) = u(2,0,0,0,...)c2 = 12c2

1c2u0

u(2,0,1,0,...) = u(2,0,0,0,...)c3 = c21c3u0

...

and so on. We proceed by the same procedure for all multi-index lengths to obtain uα in the form

uα = u0 ⊗ cα11

α1! · cα22

α2! · cα33

α3! · · · for all α = (α1, α2, α3, . . .) ∈ I,

and the form of the solution (8).It remains to prove the convergence of the solution (8) in the space X ⊗ (Sc)−1, i.e. to prove

that, for some p > 0

‖u‖2X⊗(Sc)−1,−p,c

=∑α∈I

‖uα‖2X(2Nc)−pα < ∞.

From assumption ck ≥ 1/2k, for all k ∈ N, it follows that∑

α∈I(2Nc)−pα < ∞ if p > 0. Then,for p > 3, we have

‖u‖2X⊗(Sc)−1,−p

=∑α∈I

‖u0‖2X

c2α

(α!)2(2Nc)−pα

≤ ‖u0‖2X

∑α∈I

c2α(2N)−pαc−pα

≤ ‖u0‖2X

∑α∈I

c−(p−2)α∑α∈I

(2N)−pα < ∞,

where we used property∑

α∈I a−pα < ∞, if p > 1, when a = (ak), ak ≥ 1 for all k = 1, 2, . . ..With this statement we complete the proof. �

Especially, for C = ξi , for fixed i ∈ N, Equation (7) transforms into

Du = ξi ⊗ u, Eu = u0 ∈ X. (13)

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Integral Transforms and Special Functions 105

The chaos expansion of the generalized stochastic process u ∈ X ⊗ (Sc)−1, which represents thesolution of (13), is given by

u = u0 ⊗∞∑

n=0

cni

n!Hnε(i) = u0 ⊗∞∑

n=0

1

n!Hnε(i) = u0 ⊗ exp I (ξi),

where I (·) represents the Itô integral of the Hermite function. Using the generating property ofHermite polynomials, we obtain another form of the solution

u = u0 ⊗ exp

(I (ξi) − 1

2

)= u0 ⊗ εξi

,

where εξiis the stochastic exponent of ξi .

Remark 1 In [5] it is proved that Dεh = hεh, for deterministic h. Theorem 4.1 gives a moregeneral result.

If we choose C = 0 then Equation (7) transforms to Du = 0, Eu = u0 ∈ X and has a uniquetrivial solution u = u0 in the space X ⊗ (Sc)−1.

Acknowledgements

This paper was supported by the project No. 144016, financed by the Ministry of Science, Republic of Serbia.

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