Potential Anal (2010) 32:363–387 DOI 10.1007/s11118-009-9155-3

On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle Stevan Pilipovi´c · Dora Seleši

Received: 16 April 2009 / Accepted: 10 September 2009 / Published online: 25 September 2009 © Springer Science + Business Media B.V. 2009

Abstract We treat the stochastic Dirichlet problem L♦u = h + ∇ f in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space W01,2 into the Kondratiev space (S)−1 . The operator L is assumed to be strictly elliptic in divergence form L♦u = ∇(A♦∇u + b ♦u) + c♦∇u + d♦u. Its coefficients: the elements of the matrix A and of the vectors b , c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by ♦, is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to L♦u = h + ∇ f . Keywords Generalized random process · Chaos expansion · Stochastic differential equation · Elliptic linear differential operator · Generalized expectation · Wick product Mathematics Subject Classifications (2000) 60H30 · 60G20 · 46N30 · 46F30 · 46E35 · 35J05

S. Pilipovi´c · D. Seleši (B) Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovi´ca 4, 21000 Novi Sad, Serbia e-mail: [email protected] S. Pilipovi´c e-mail: [email protected]

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1 Introduction Consider the stochastic Dirichlet problem L u(x, ω) = h(x, ω) +

n 

Di f i (x, ω),

x ∈ I, ω ∈ ,

i=1

u(x, ω) ∂ I = g(x, ω),

(1)

where I ⊂ Rn is an open bounded set, (, F , P) is a probability space, h, g and f i , i = 1, 2 . . . n are generalized random processes defined as in [11] and L is a linear elliptic operator of the form L u(x, ·) =

n 

⎛ ⎞ n  Di ⎝ aij(x, ·)Dju(x, ·) + b i (x, ·)u(x, ·)⎠

i=1

+

n 

j=1

c i (x, ·)Di u(x, ·) + d(x, ·)u(x, ·)

(2)

i=1

Following [4], where the deterministic equation of this form was given a physical meaning, our stochastic version Eq. 1 describes a diffusion process in a stochastic anisotropic medium, with transport and creation also dependent on some random factors, and with a stochastic boundary value. We will assume that the coefficients of L are essentially bounded processes, that the boundary process belongs to an appropriate Sobolev-type space, and that h and f i are square–integrable in the x-variable. The fact that L in Eq. 2 is given in divergence form, will make it suitable to work with in Sobolev spaces in terms of weak derivatives, and using Sobolev space methods we are able to solve the problem under the assumptions that f i , i = 1, . . . , n are square–integrable (which is of course less than requiring that f i belong to the Sobolev space). The case when the coefficients of L are measurable deterministic functions, or deterministic Colombeau generalized functions, was treated in our previous paper [12]. However, if the coefficients of the operator L are also generalized random processes, then a further problem arises: How to interpret the product between two generalized random processes? In [14] product interpreted as the Wick  n was the n ij product, and a solution to the equation i=1 Di j=1 a (x, ·)♦ Dj u(x, ·) = h(x, ·) was found as an element of the Sobolev-Kondratiev space. A similar approach can be found in [2], where the tensor product of the Sobolev space and the space of stochastic trigonometric functions was used, and the product was taken pointwisely. In [3] the author uses the direct product of the classical Sobolev space and a generalized Sobolev space as a probability space, which allows the product to be interpreted as ordinary product of two functions. Our approach will follow the deterministic theory of elliptic PDEs given in Section 8 of Gilbarg and Trudinger’s monograph [6]. Using the nuclearity of the Kondratiev spaces (S)1 and (S)−1 , we are able to develop a framework of generalized stochastic process spaces endowed with the concepts of strong and weak differentiation, that allows us to interpret problem (1) in a way which enables us to use the

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365

techniques of [6]. We will introduce two concepts of solutions: a generalized weak solution and a componentwise solution. In this part of the paper we prove as the main theorem the stochastic weak maximum principle: If L♦u is positive in a weak sense, then u attains its supremum on the boundary of I. This will imply that the homogeneous Dirichlet problem L♦u = 0 has only a trivial solution. In the second part of this article we will discuss the existence and uniqueness problems of the nonhomogeneous Dirichlet problem L♦u = h + ∇ f . The paper is organized in the following manner: In Section 2 we provide a basic overview of the notation and recall on some results that are needed for further reading. In Section 3 we develop all necessary tools for the Hilbert space methods to be used, especially we explain the notions of weak and strong derivatives of generalized random processes and provide some estimates on the Wick product. Section 4 is devoted to the Dirichlet problem (1), when the coefficients of the operator L are generalized random processes. We will give an interpretation of the operator itself, of the equation and its solution in terms of Wick products. We seek for  a weak solution of Eq. 1 in the Hilbert space of generalized random processes L W01,2 (I ), (S)−1 , where W01,2 (I ) denotes the Sobolev space and (S)−1 denotes the Kondratiev space.

2 Preliminaries Throughout the paper I will denote an open bounded subset of Rn . Let W k,2 (I ), W0k,2 (I ) and W −k,2 (I ), k ∈ N0 , N0 = N ∪ {0}, be the usual notations for the Sobolev spaces (see [1]) and C0k (I ) for the space of k times differentiable functions with in direction compact support. We will use the notation Di for the weak derivative n of the ith coordinate in Rn , D for the total differential Du = i=1 Di u, and Dα = Dα1 1 Dα2 2 · · · Dαnn for the αth differential. The dual pairing between W −k,2 (I ) and W0k,2 (I ) will be denoted by ·, ·W . In case of the pairing of W k,2 (I ) and W0k,2 (I ), this reduces to the scalar product ·, · L2 . Let the basic probability space (, F , P) be (S  (Rn ), B , μ), where S  (Rn ) denotes the space of tempered distributions, B the sigma-algebra generated by weak topology and μ denotes the white noise measure given by the Bochner-Minlos theorem (cf. [5, 9, 10]). Let (L)2 = L2 (S  (Rn ), B , μ), and Hα , α ∈ I be the Fourier–Hermite orthogonal basis of (L)2 , where I denotes the set of sequences of integers which have only finitely many nonzero Let

 components. γj − pγ (2N)γ = ∞ (2 j ) , where γ = (γ , γ , . . .) ∈ I . It is known that (2 N ) < ∞ 1 2 γ ∈I j=1 if p > 1. The space of the Kondratiev stochastic test functions (S)1 = p∈N0 (S)1, p is the projective limit of the spaces (S)1, p =

f=



cα Hα ∈ (L) , cα ∈ R :

2

α∈I

f 21, p =



c2α (α!)2 (2N) pα < ∞

The space of the Kondratiev stochastic generalized functions (S)−1 = is the inductive limit of the spaces (S)−1,− p =

F=

, p ∈ N0 .

α∈I

 α∈I

b α Hα , b α ∈ R :

F 2−1,− p =

 α∈I



p∈N0 (S)−1,− p

b 2α (2N)− pα < ∞

, p ∈ N0 .

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 The action of F onto a test function f is given by the dual pairing F, f  S = α∈I α!b α cα . The generalized expectation of F ∈ (S)−1 is defined by E(F ) = F, 1 S = b 0 . We  will also consider the scalar product in (S)−1,− p given by F, f −1,− p = α∈I b α cα (2N)− pα . We consider generalized random processes (GRPs) as linear continuous mappings from the Sobolev space W0k,2 (I ) into the space of Kondratiev generalized random variables (S)−1 and use the notation WS ∗k = L(W0k,2 (I ), (S)−1 ). Especially, for k = 1 we denote the space of GRPs by  WS ∗ = L W01,2 (I ), (S)−1 . Since (S)1 is a nuclear space and W −k,2 (I ) is a Frèchet space,  we may regard GRPs also as elements of tensor product spaces (cf. [13]): L W0k,2 (I ), (S)−1 ∼ =    k,2 k,2 L (S)1 , W −k,2 (I ) ∼ = Bil W0 (I ) × (S)1 , R ∼ = L W0 (I ) ⊗ (S)1 , R ∼ = W −k,2 (I ) ⊗ (S)−1 . Here the notation ⊗ can be interpreted both as the π –completion and as the ε–completion of the tensor product spaces, since they are the same by the nuclearity of (S)1 . Especially, WS ∗ is the dual of W0 S = W01,2 (I ) ⊗ (S)1 . We state our main result from [11], which was used also in [12]: Theorem 1 Following conditions are equivalent:  (i) u ∈ L W0k,2 (I ), (S)−1 . (ii) u can be represented in the form u(x, ω) =



fα (x) ⊗ Hα (ω),

x ∈ I, ω ∈ , fα ∈ W −k,2 (I ), α ∈ I

(3)

α∈I

and there exists p ∈ N0 such that for each bounded set B ⊂ W k,2 (I ) sup ||u(x, ω), ϕ(x)W ||2−1,− p = sup

ϕ∈B

ϕ∈B



| fα , ϕW |2 (2N)− pα < ∞.

(4)

α∈I

(iii) u can be represented in the form Eq. 3 and there exists p ∈ N0 such that ||u(x, ω)||2W −k,2 ⊗(S )−1,− p =



fα 2W −k,2 (2N)− pα < ∞.

(5)

α∈I

We will also use the Hilbert structure of (S)−1,− p for fixed p ∈ N0 and consider the spaces W01,2 (I ) ⊗ (S)−1,− p and W −1,2 (I ) ⊗ (S)−1,− p as its S-antidual. (S)−1,− p is the antidual of itself through the scalar product ·, ·−1,− p . We will use the notation L2 (I ) ⊗ (S)−1,− p for the space of elements of the form Eq. 3 with fα ∈ L2 (I ), α ∈ I , such that (5) holds with fα 2L2 instead of fα 2W −k,2 . In a similar way we define Banach spaces H(I ) ⊗ (S)−1,− p , for H(I ) = L p (I ), p ∈ [1, ∞), H(I ) = W0k,2 (I ), H(I ) = W k,2 (I ), k ∈ N0 . Clearly, if H  (I ) is the dual of H(I ), then H  (I ) ⊗ (S)−1,− p is the dual of H(I ) ⊗ (S)1, p . Since this paper is related to solving Eq. 1 we will use the Hilbert structure of (S)−1,− p and consider H  (I ) ⊗ (S)−1,− p as the S-antidual of H(I ) ⊗ (S)−1,− p .

The Stochastic Weak Maximum Principle

367

If F ∈ W −1,2 (I ) ⊗ (S)−1,− p and f ∈ W01,2 (I ) ⊗ (S)−1,− p , then we will use the notation  Fα , fα W (2N)− pα , (6) F, f  = α∈I

  where F = α∈I Fα (x) ⊗ Hα (ω), Fα ∈ W −1,2 (I ) and f = α∈I fα (x) ⊗ Hα (ω), fα ∈ W01,2 (I ). In case that F, f ∈ L2 (I ) ⊗ (S)−1,− p , then Eq. 6 reduces to F, f  =



Fα , fα  L2 (2N)− pα =

α∈I

 α∈I

F(x) f (x)dx(2N)− pα .

(7)

I

This scalar product should be distinguished from the dual pairing of W −1,2 (I ) ⊗ (S)−1 and W01,2 (I ) ⊗ (S)1 given by F, f W S =



α!Fα , fα W .

(8)

α∈I

The product between two generalized random processes in Eq. 2 will be interpreted as the Wick product; for this purpose we recall the definition of the Wick product in the Kondratiev  spaces (see also [8]). LetF, G ∈ (S)−1 be given by their chaos expansions F(ω) = α∈I fα Hα (ω), G(ω) = β∈I gβ Hβ (ω), fα , gβ ∈ R. The Wick product of F and G is the unique element in (S)−1 defined by: F ♦G(ω) =

 γ ∈I

⎛ ⎝



⎞ fα gβ ⎠ Hγ (ω).

α+β=γ

3 Derivatives and Wick Product Estimates 3.1 Differentiation of GRPs Let C(I ) be the space of bounded continuous functions on I endowed with the supremum norm and let Ck (I ), k ∈ N, denote its subspace consisting of functions f that have bounded continuous derivatives up to the order k supplied with the norm

f Ck = supx∈I,|ν|≤k |Dν f (x)|. Definition 1 Let Ck (I; (S)−1 ) denote the space of generalized random processes  F(x, ω) = α∈I fα (x)Hα (ω), x ∈ I, ω ∈ , for which fα ∈ Ck (I ), α ∈ I , and there exists p ∈ N0 such that 

2 − pα

fα C < ∞. k (2N)

(9)

α∈I

By the nuclearity of (S)−1 , we may also consider Ck (I; (S)−1 ) as the tensor product space Ck (I ) ⊗ (S)−1 .

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Now we introduce the concepts of strong and weak derivatives of generalized random processes. We will restrict our attention to the first order derivatives in direction ei ∈ Rn , ei = (0, 0, . . . , 1, 0 . . . , 0), since only these are needed for solving the Dirichlet problem, but one can easily define higher order derivatives, too. Definition 2 Let F ∈ C(I ) ⊗ (S)−1 , respectively F ∈ C(I ) ⊗ (S)1 . If for ei ∈ Rn , ei = (0, 0, . . . , 1, 0 . . . , 0) there exists a process G ∈ C(I ) ⊗ (S)−1 , respectively G ∈ C(I ) ⊗ (S)1 , such that for some p ∈ N0 , respectively every p ∈ N0 , and every x ∈ I:  2  F(x + hei , ω) − F(x, ω)   − G(x, ω) = 0, lim   h→0 h −1,− p

(10)

 2   F(x + hei , ω) − F(x, ω)  − G(x, ω) lim  = 0, h→0  h 1, p

(11)

respectively

then we say that F is differentiable in direction ei , we call G the ith (strong) derivative of F and denote it as Di F = G. Definition 3 Let k ∈ N0 . For F ∈ W k,2 (I ) ⊗ (S)−1 , respectively F ∈ W k,2 (I ) ⊗ (S)−1,− p , its weak derivative is defined by Di F, φW S = −F, Di φW S ,

φ ∈ C0k+1 (I ) ⊗ (S)1 ,

(12)

respectively Di F, φ = −F, Di φ,

φ ∈ C0k+1 (I ) ⊗ (S)−1,− p ,

(13)

and Di φ is taken in the sense of the previous definition. Note that Eq. 12 is equivalent to requiring that  2   F(x + he , ω) − F(x, ω)  i   − Di F(x, ω), ϕ(x) lim   h→0  h L2 (I ) 

= 0,

(14)

1, p

for all ϕ ∈ C0k+1 (I ), and respectively for Eq. 13 the norm in Eq. 14 is to be replaced by · −1,− p . Also note that C0k+1 (I ) ⊗ (S)−1 is dense in W0k+1,2 (I ) ⊗ (S)−1 , which will be used to define weak derivatives in W −k,2 (I ) ⊗ (S)−1 . Moreover, C0k+1 (I ) ⊗ (S)−1,− p is dense in W0k+1,2 (I ) ⊗ (S)−1,− p for fixed p ∈ N0 , so this will be used to define weak derivatives in W −k,2 (I ) ⊗ (S)−1,− p . The following lemma ensures that the derivative can be taken componentwise in the chaos expansion.

The Stochastic Weak Maximum Principle

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Lemma 1 (a) Let F ∈ C(I ) ⊗ (S)−1 , be given by F(x, ω) = α ∈ I.



α∈I

fα (x) ⊗ Hα (ω), fα ∈ C(I ),

 (i) Let G = Di F exist and be given by G(x, ω) = α∈I gα (x) ⊗ Hα (ω), gα ∈ ∈ I , and F ∈ C1 (I ) ⊗ (S)−1 . C(I ). Then, gα (x) = Di fα (x) for every α  (ii) If Di fα (x) ∈ C(I ) for every α ∈ I , and if α∈I |Di fα (x)|2 (2N)−qα < ∞ for some q ∈ N0 , then Di F exists in C(I ) ⊗ (S)−1 and Di F(x, ω) =



Di fα (x) ⊗ Hα (ω).

(15)

α∈I

(b) Let F ∈ C(I ) ⊗ (S)1 , be given by F(x, ω) = α ∈ I.



α∈I

fα (x) ⊗ Hα (ω), fα ∈ C(I ),

 (i) Let G = Di F exist and be given by G(x, ω) = α∈I gα (x) ⊗ Hα (ω), gα ∈ C(I ). Then, gα (x) = Di fα (x) for every α  ∈ I , and F ∈ C1 (I ) ⊗ (S)1 . (ii) If Di fα (x) ∈ C(I ) for every α ∈ I , and if α∈I |Di fα (x)|2 (2N)−qα < ∞ for every q ∈ N0 , then Di F exists in C(I ) ⊗ (S)1 and Eq. 15 holds.

Proof We will prove only part (a), and the proof for (b) is similar. Let r ∈ N0 be such that F ∈ C(I ) ⊗ (S)−1,−r . (i) From 2    fα (x + hei ) − fα (x)   lim − gα (x) (2N)− pα = 0,  h→0 h α∈I

for some p ∈ N0 , follows that for every ε > 0 there exists δ(ε) > 0 such that  2   fα (x+hei )− fα (x)  |h| < δ implies − gα (x) (2N)− pα < ε. Therefore, for all α∈I  h α ∈ I, 2    fα (x + hei ) − fα (x)  < ε(2N) pα .  (x) − g α   h Thus, for all ε > 0 and for all α ∈ I , Di fα (x) exists and |Di fα (x) − gα (x)|2 < ε(2N) pα . Now,     Di fα (x) − gα (x)2 (2N)−( p+2)α < ε (2N)−2α = Cε. α∈I

α∈I

 2 −qα = 0, for q = p + 2, From this follows that α∈I |Di fα (x) − gα (x)| (2N) i.e. Di fα = gα , for all α ∈ I . By the assumption fα ∈ C(I ) and gα ∈ C(I ), and since gα = Di fα , we have that fα ∈ C1 (I ) for all α ∈ I . Clearly,  2 −sα < ∞ for s = max{ p, q, r}. α∈I fα C1 (2N)

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 (ii) We have to prove that G(x, ω) = α∈I Di fα (x) ⊗ Hα (ω) satisfies condition (10). Indeed, for p = max{q, r}, we have that 2    F(x + hei , ω) − F(x, ω)  lim − G(x, ω)  h→0  h −1,− p  2   fα (x + hei ) − fα (x)  − Di fα (x) (2N)− pα = 0. ≤ lim  h→0 h α∈I

  Lemma 2 (a) Let F ∈ W k,2 (I )⊗(S)−1 be given by F(x, ω) =



α∈I

fα (x)⊗ Hα (ω), fα ∈ W k,2 (I ).

(i) Let the weak derivative G = Di F exist  in the sense of the dual pairing ·, ·W S and be given by G(x, ω) = α∈I gα (x) ⊗ Hα (ω), gα ∈ W k−1,2 (I ). Then, gα (x) = Di fα (x) for every α ∈ I . (ii) If the Di fα (x) exist in ∈ W k−1 (I ) for every α ∈ I , and  weak derivatives 2 −qα if < ∞ for some q ∈ N0 , then Di F exists in α∈I ||Di fα ||W k−1 (2N) k−1,2 (I ) ⊗ (S)−1 in the sense of the dual pairing and W  Di fα (x) ⊗ Hα (ω). (16) Di F(x, ω) = α∈I

(b) Let F ∈ W k,2 (I ) ⊗ (S)−1,− p be of the form as in (a). (i) Let the weak derivative G = Di F exist  in the sense of S-antidual pairing ·, · and be given by G(x, ω) = α∈I gα (x) ⊗ Hα (ω), gα ∈ W k−1,2 (I ). Then, gα (x) = Di fα (x) for every α ∈ I . (ii) If Di fα (x) exist in ∈ W k−1 (I ) for every α ∈ I , and if the weak derivatives 2 − pα < ∞, then Di F exists in W k−1,2 (I ) ⊗ (S)−1,− p α∈I ||Di fα ||W k−1 (2N) in the sense of S-antidual pairing and Eq. 16 holds. Proof We will prove only part (b). For part (a) one only has to replace ·, · by ·, ·W S .  (i) From Lemma 1 we obtain that for all φ = α∈I φα (x) ⊗ Hα (ω) ∈ C0k+1 (I ) ⊗ (S)−1,− p ,     Di F, φ = −F, Di φ = − Fα ⊗ Hα , Di φα ⊗ Hα α∈I

α∈I

  =− Fα , Di φα  L2 (2N)− pα = Di Fα , φα  L2 (2N)− pα α∈I

=

 



α∈I

Di Fα ⊗ Hα , φ .

α∈I

 Thus, Di F = α∈I Di Fα ⊗ Hα . (ii) This can be shown in an analogous way.

 

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Now we introduce weak derivatives in the dual spaces W −k,2 (I ) ⊗ (S)−1 and W (I ) ⊗ (S)−1,− p . −k,2

Definition 4 Let k, p ∈ N0 and let F ∈ W −k,2 (I )⊗(S)−1 , respectively F ∈ W −k,2 (I ) ⊗ (S)−1,− p . The ith weak derivative of F, denoted by Di F is given by its dual action Di F, vW S = −F, Di vW S ,

for all v ∈ W0k+1,2 (I ) ⊗ (S)1 ,

(17)

respectively Di F, v = −F, Di v,

for all v ∈ W0k+1,2 (I ) ⊗ (S)−1,− p .

(18)

Note that Di v is also a weak derivative taken in the sense of Definition 3. Since C0k+1 (I ) ⊗ (S)1 is dense in W0k+1,2 (I ) ⊗ (S)1 , respectively C0k+1 (I ) ⊗ (S)−1,− p is dense in W0k+1,2 (I ) ⊗ (S)−1,− p it is enough to consider v ∈ C0k+1 (I ) ⊗ (S)−1,− p in Eq. 17, respectively in Eq. 18. Lemma 3 Let F ∈ W −k,2 (I ) ⊗ (S)−1 , respectively F ∈ W −k,2 (I ) ⊗ (S)−1,− p be given by F(x, ω) = α∈I fα (x) ⊗ Hα (ω), fα ∈ W −k,2 (I ). Then its weak derivative Di F is the unique element of W −k−1,2 (I ) ⊗ (S)−1 , respectively W −k−1,2 (I ) ⊗ (S)−1,− p , given by the expansion  Di fα (x) ⊗ Hα (ω), Di F(x, ω) = α∈I

where Di fα (x), α ∈ I , denotes the weak derivative in the Sobolev space W −k−1,2 (I ). Proof We will consider only the second case i.e. interpret the S-antidual pairing  and prove that α∈I Di fα (x) ⊗ Hα (ω) satisfies Definition 4. Let v ∈ W0k+1,2 (I ) ⊗ (S)−1,− p be of the form v(x, ω) = ϕ(x)θ(ω), ϕ ∈ W0k+1,2 (I ), θ ∈ (S)−1,− p . Since Di fα are weak derivatives in Sobolev sense, we have that Di fα , φW = − fα , Di φW , α ∈ I , for all φ ∈ W0k+1,2 (I ). Thus, 

 α∈I

 Di fα ⊗ Hα , v =



Di fα , ϕW Hα , θ−1,− p = −

α∈I

 =−



 fα , Di ϕW Hα , θ−1,− p

α∈I

 α∈I

 fα ⊗ Hα , (Di ϕ)θ =





 fα ⊗ Hα , Di v .

α∈I

  3.2 Essentially Bounded GRPs We now define the Wick product of two generalized random processes (recall Theorem 1) in an analogous way as it was defined in the Kondratiev spaces. First we introduce the class of essentially bounded (in the x variable) generalized random processes, for which this product will be well defined.

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Definition 5 Let L∞ (I; (S)−1,− p ), p ∈ N0 , constitute of elements F ∈ WS ∗ for which essup F(x, ω) −1,− p < ∞.

(19)

x∈I

The space of essentially bounded generalized random processes, denoted by  L∞ (I; (S)−1 ) is L∞ (I; (S)−1 ) = p∈N0 L∞ (I; (S)−1,− p ) endowed with the inductive limit topology. Lemma 4  Let F ∈ WS ∗ be a generalized random process given by its chaos expansion F(x, ω) = α∈I fα (x) ⊗ Hα (ω). The following conditions are equivalent: (i) F ∈ L∞ (I; (S)−1 ). (ii) For all α ∈ I , fα ∈ L∞ (I ) and there exists q ∈ N0 such that 

fα 2L∞ (I ) (2N)−qα < ∞.

(20)

α∈I

(iii) For all α ∈ I , fα ∈ L∞ (I ) and C(ω) =



α∈I

fα L∞ (I ) Hα (ω) ∈ (S)−1 .

Proof It is clear that conditions (ii) and (iii) are equivalent. Also,       2 2 − pα  | fα (x)| (2N)  ≤

fα 2L∞ (I ) (2N)− pα , essup F(x, ω) −1,− p =    x∈I α∈I

L∞ (I )

α∈I

and thus Eq. 20 implies Eq. 19. The converse is a consequence of the nuclearity  2 ˜ of (S) : Assume that M = essup | f (x)| (2N)− pα < ∞, i.e. that F(x) = −1 α x∈I α∈I  2 − pα ∞ ∞ ∞ the dual pairing of L | f (x)| (2 N ) ∈ L (I ). Denote by ·, · (I ) and α L α∈I ˜ ˜ L∞ (I ) ϕ L1 (I ) , i.e. L1 (I ). Thus, for every ϕ ∈ L1 (I ), we have | F(x), ϕ(x) L∞ | ≤ F

   fα , ϕ L∞ 2 (2N)− pα ≤ M ϕ L1 (I ) . α∈I

From this follows that for every α ∈ I , | fα , ϕ L∞ |2 (2N)− pα ≤ M ϕ L1 (I ) , and thus

fα 2L∞ (I ) (2N)− pα ≤ M,

for all α ∈ I .

Cleary, fα 2L∞ (I ) ≤ M(2N) pα < ∞, for all α ∈ I . Let q = p + 2. Then,  

fα 2L∞ (I ) (2N)−qα ≤ M (2N)−2α < ∞. α∈I

α∈I

  From the previous lemma it follows that L∞ (I; (S)−1 ) ∼ = L∞ (I )⊗(S)−1 . Since ∞ L (I ) is complete and (S)1 is nuclear, we also have L (I ) ⊗ (S)−1 ∼ = L(L1 (I ); (S)−1 ). Clearly, for fixed p ∈ N0 the following holds:   L∞ (I; (S)−1,− p ) ⊇ L∞ I ⊗ (S)−1,− p ∼ = L L1 (I ); (S)−1,− p . 1

The Stochastic Weak Maximum Principle

373

Definition 6 The spaces L p (I; (S)−1 ), p ≥ 1, consist of GRPs F(x, ω) = Hα (ω) such that  1p  p

F(x, ω) −1,−r dx <∞



α∈I

fα (x)⊗

I

for some r ∈ N0 . It is easy to show that this is equivalent to the fact that for each α ∈ I , fα ∈ L p (I ), and 

fα 2L p (I ) (2N)−(r+2)α < ∞. α∈I

Thus, L p (I; (S)−1 ) ∼ = L p (I ) ⊗ (S)−1 . Definition 7 Let m ∈ N. Denote by W m,∞ (I; (S)−1 ) the space of GRPs F such that Dα F ∈ L∞ (I; (S)−1 ) for 0 ≤ |α| ≤ m. As above, we note that W m,∞ (I; (S)−1 ) ∼ = W m,∞ (I ) ⊗ (S)−1 as a consequence of the nuclear structure of  (S)−1 . Since H(I )⊗(S)−1 = r∈N0 H⊗(S)−1,−r for any of the spaces H(I ) = L p (I ), p ≥ 1, or H(I ) = W0k,2 (I ), H(I ) = W k,2 (I ), k ∈ N0 , and H(I ) ⊗ (S)−1,−r ⊂ H ⊗ (S)−1,−q if r < q, we will consider in the sequel H(I ) ⊗ (S)−1,−r with its S-antidual H  (I ) ⊗ (S)−1,−r for fixed r ∈ N0 . 3.3 The Mapping Kq   2 Let q ∈ N0 and F = α∈I fα (x)⊗ Hα (ω) ∈ L∞ (I )⊗(S)−1,−(q−4) . Thus, α∈I fα L∞ 2 −(q−4)α −(q−4)α (2N) < ∞, which implies that there exists M > 0 such that fα L∞ (2N) ≤ M for each α ∈ I . This implies further that 

q

fα L∞ (2N)− 2 α =

α∈I



fα L∞ (2N)−(( 2 −2)+2)α ≤ q

α∈I

√

M



(2N)−2α < ∞.

α∈I

  Thus, if we put F˜ = α∈I | fα (x)| ⊗ Hα (ω), then F˜ ∈ L∞ (I ) ⊗ (S)−1,− q2 and  − q2 α ˜ 2∞ .

F

α∈I fα L∞ (I ) (2N) L ⊗(S )−1,−q/2 = We denote by Kq the mapping Kq : L∞ (I ) ⊗ (S)−1,−(q−4) → L∞ (I ) ⊗ (S)−1,− q2 defined by  | fα (x)| ⊗ Hα (ω). (21) Kq (F ) = F˜ = α∈I

It is a continuous mapping from L∞ (I ) ⊗ (S)−1,−(q−4) to L∞ (I ) ⊗ (S)−1,− q2 . A similar consideration can be carried out for the space W 1,∞ (I ) ⊗ (S)−1 :   q

fα 2W 1,∞ (2N)−(q−4)α < ∞, then

fα W 1,∞ (2N)− 2 α < ∞. If α∈I

α∈I

(22)

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S. Pilipovi´c, D. Seleši

 ˜ Example 1 Singular white noise W(x, ω) = ∞ n=1 ξn (x)Hεn (ω) and W(x, ω) = ∞ √ ∞ |ξn (x)|Hεn (ω) are elements of L (I; (S)−1 ). This follows from the fact that n=1 1 the Hermite functions are uniformly bounded i.e. |ξn (x)| ≤ const · π − 4 , n ∈ N. 3.4 Wick Products Definition 8 Let F ∈ W 1,∞ (I; (S)−1,−q+4 ) for some q ∈ N 0 be given by F(x, ω) =   f (x) ⊗ Hα (ω) and G ∈ WS ∗ be given by G(ω) = β∈I gβ (x) ⊗ Hβ (ω). The α∈I α Wick product of F and G is the unique element in WS ∗ defined by: ⎛ ⎞   ⎝ F ♦G(x, ω) = fα (x)gβ (x)⎠ ⊗ Hγ (ω). γ ∈I

α+β=γ

The next lemma shows that the Wick product is well defined, and that for fixed F the mapping G  → F ♦G is continuous. Lemma 5 If F ∈ W 1,∞ (I )⊗(S)−1,−( p−4) and if G ∈ W −1,2 (I )⊗(S)−1,− p , then F ♦G ∈ W −1,2 (I ) ⊗ (S)−1,− p . Moreover, there exists C > 0 such that:

F ♦G W −1,2 (I )⊗(S )−1,− p ≤ C G W −1,2 (I )⊗(S )−1,− p . Proof We will use the known fact from Sobolev spaces, that the multiplication W01,2 (I ) × W 1,∞ (I )  (ϕ, ψ) → ϕ · ψ ∈ W 1,2 (I ) is continuous, thus multiplication of elements of W 1,∞ (I ) and W −1,2 (I ) is well defined. Also, we use the estimate

aα b β W −1,2 ≤ M aα W 1,∞ b β W −1,2 . We have ⎛ ⎞12   2 − pγ ⎠

F ♦G W −1,2 (I )⊗(S )−1,− p = ⎝

Fα (x)Gβ (x) −1,2 (2N) W

γ ∈I

α+β=γ

(I )

⎛

⎞12   =⎝

aα (x)b γ −α (x) 2W −1,2 (I ) ⎠ , γ ∈I

α∈I

p

p

where aα (x) = Fα (x)(2N)− 2 α , b β (x) = Gβ (x)(2N)− 2 β , and we use the convention b γ −α = 0 whenever the multiindex γ − α has negative components. Now, applying the generalized Hölder inequality ⎞12 ⎞12 ⎛      ⎝

aα b γ −α 2W −1,2 (I ) ⎠ ≤ M

aα W 1,∞ (I ) ⎝

b β 2W −1,2 (I ) ⎠ ⎛

γ ∈I

α∈I

α∈I

β∈I

we obtain 

F ♦G 2W −1,2 ⊗(S )−1,− p ≤ M

 α∈I

Fα W 1,∞ (I ) (2N)−

p 2α

2 ⎛  ⎝

Gβ 2 β∈I

W −1,2 (I ) (2N)

⎞ − pβ ⎠

.

The Stochastic Weak Maximum Principle

Now by Eq. 22 we have that  = we obtain the estimate

375



p

α∈I

Fα W 1,∞ (I ) (2N)− 2 α < ∞, thus for C = M2

F ♦G W −1,2 (I )⊗(S )−1,− p ≤ C G W −1,2 (I )⊗(S )−1,− p .   For the Dirichlet problem (1), we will also need multiplication in L2 (I; (S)−1 ) and L∞ (I; (S)−1 ). Lemma 6 If F ∈ L∞ (I ) ⊗ (S)−1,−( p−4) and if G ∈ L2 (I ) ⊗ (S)−1,− p , then F ♦G ∈ L2 (I ) ⊗ (S)−1,− p . Moreover, there exists C > 0 such that: ˜ 2L∞ (I )⊗(S )

F ♦G L2 (I )⊗(S )−1,− p ≤ C F

p −1,− 2

G L2 (I )⊗(S )−1,− p .

Proof The proof is similar to that of the previous lemma, one just has to use the fact that the product of a square–integrable function and an essentially bounded function is again square–integrable, and apply the estimate fα gβ L2 ≤ C fα L∞ gβ L2 . Thus, ⎞ 2 ⎛    p

Fα L∞ (I ) (2N)− 2 α ⎝

Gβ 2L2 (I ) (2N)− pβ ⎠

F ♦G 2L2 ⊗(S )−1,− p ≤ C α∈I

and now if we put F˜ =



α∈I

β∈I

√

|Fα | ⊗ Hα , then we obtain by Eq. 21 that

˜ 2L∞ (I )⊗(S )

F ♦G L2 (I )⊗(S )−1,− p ≤ C F

p −1,− 2

G L2 (I )⊗(S )−1,− p .  

Example 2 Let  W(x, ω) denote singular white noise from Example 1. Then, 1 ♦n (x, ω) is also an element of L∞ (I; (S) ). Indeed, applyexp♦ W(x, ω) := ∞ −1 n=0 n! W ♦(n−1)

˜ 2 ˜ 2 ing Lemma 6 we obtain W ♦n −1,− p ≤ C W

−1,− p ≤ C W −1,− p/2 −1,− p/2 W 2n 2 ♦ (n−2) n ˜ ˜ C W

−1,− p ≤ · · · ≤ C W

−1,− p/2 W −1,− p/2 , and thus  ∞ ∞    1  1      ♦n ♦n ≤ W (x, ω)  W (x, ω) ∞   L ⊗(S )−1,− p n! n! ∞ n=0 n=0 L ⊗(S )−1,− p

∞ 2n  Cn  ˜  W(x, ω) ∞ L ⊗(S )−1,− p/2 n! n=0   ˜ = exp C W(x, ω) 2L∞ ⊗(S )−1,− p/2 < ∞.

≤

4 The Dirichlet Problem with Stochastic Coefficients In order to prove existence and uniqueness of the solution of Eq. 1 provided the coefficients of the operator L are generalized random processes, we have to go

376

S. Pilipovi´c, D. Seleši

deeper into the Hilbert space structure of L(W01,2 , (S)−1,− p ) than we did in [12]. The main reference for this approach are [6] and [4]. Recall that ·, · denotes the scalar product on L2 (I ) ⊗ (S)−1,− p given in Eq. 7. In fact, this p can be chosen to be p = max

1≤i, j≤n

 ij  k + 4, li + 4, zi + 4, s + 4, t, q, mi ,

(23)

where  kij = min n ∈ N0 :  li = min n ∈ N0 :  zi = min n ∈ N0 :  s = min n ∈ N0 :  t = min n ∈ N0 :  q = min n ∈ N0 :  mi = min n ∈ N0 :

 a˜ ij ∈ L∞ (I ) ⊗ (S)−1,−n ,  b˜ i ∈ L∞ (I ) ⊗ (S)−1,−n ,  c˜i ∈ L∞ (I ) ⊗ (S)−1,−n ,  d˜ ∈ L∞ (I ) ⊗ (S)−1,−n ,  h ∈ L2 (I ) ⊗ (S)−1,−n ,  g ∈ W 1,2 (I ) ⊗ (S)−1,−n ,  f i ∈ L2 (I ) ⊗ (S)−1,−n ,

 and a˜ ij = Kq aij , b˜ i = Kq (b i ), c˜ i = Kq (c i ), d˜ = Kq (d ), where aij, b i , c i , d, h, g, f i are given in Eqs. 1 and 2, and the mapping Kq is defined in Eq. 21. From now on we will assume that p stands for this p given in Eq. 23 and we will work in the Hilbert space (S)−1,− p . 4.1 Interpretation of the Operator L According to Definition 5, we will assume (similarly as in the deterministic case) that the coefficients aij, b i , ci , d, for i, j = 1, 2 . . . , n, of the operator L are essentially bounded GRPs, and will thus interpret the action of L onto a GRP u ∈ WS ∗ as:

L♦u(x, ω) =

n 

⎛ Di ⎝

i=1

+

n 

n 

⎞ aij(x, ω)♦ Dju(x, ω) + b i (x, ω)♦u(x, ω)⎠

j=1

c i (x, ω)♦ Di u(x, ω) + d(x, ω)♦u(x, ω).

(24)

i=1

Now the operator L acts as a differential operator (in the x variable) and as a Wick–multiplication operator (in the ω variable) as well. Thus, it is natural to consider the “package” L♦ as a whole unit. But now we have to take into account that for Wick multiplication in general  f ♦g, h  =  f, g♦h, while in the deterministic case we always have  fg, h L2 =  f, gh L2 for ordinary multiplication by a test function g. Nevertheless, we will develop the necessary tools for Wick calculations.

The Stochastic Weak Maximum Principle

377

First we note that Wick multiplication satisfies the chain rule: Lemma 7 (i) For f ∈ W 2,∞ (I; (S)−1 ) and for g ∈ WS ∗ we have Di ( f ♦g) ∈ WS ∗ and Di ( f ♦g) = (Di f )♦g + f ♦(Di g), (ii) For f ∈ W (I; (S)−1 ) and for g ∈ W L2 (I; (S)−1 ) and Eq. 25 holds. 1,∞

1,2

i = 1, 2, . . . , n.

(25)

(I; (S)−1 ) we have Di ( f ♦g) ∈

Proof

  (i) Let f (x, ω) = α fα (x) ⊗ Hα (ω), g(x, ω) = β gβ (x) ⊗ Hβ (ω). Then, for arbitrary i = 1, 2, . . . , n (recall, Di denotes the weak derivative w.r.t. xi ), we have   Di fα (x) ⊗ Hα (ω), and Di g(x, ω) = Di gβ (x) ⊗ Hβ (ω). Di f (x, ω) = α∈I

β∈I

Due to definition of Wick multiplication, f ♦g(x, ω) = gβ (x)) ⊗ Hγ (ω), and thus ⎞ ⎛   Di ⎝ fα (x)gβ (x)⎠ ⊗ Hγ (ω) Di f ♦g(x, ω) = γ ∈I

=

 γ ∈I

=

 γ ∈I

⎛ ⎝

α+β=γ





⎝



 α+β=γ

fα (x)

⎞

Di [ fα (x)gβ (x)]⎠ ⊗ Hγ (ω)

α+β=γ

⎛

γ ∈I (

⎞ Di fα (x)gβ (x) + fα (x)Di gβ (x)⎠ ⊗ Hγ (ω)

α+β=γ

= Di f (x, ω)♦g(x, ω) + f (x, ω)♦ Di g(x, ω). Since f ∈ W 2,∞ (I; (S)−1 ), we have that Di f ∈ W 1,∞ (I; (S)−1 ), and thus all products are well defined (Lemma 5). (ii) Now we use the fact that f, Di f ∈ L∞ (I; (S)−1 ) and g, Di g ∈ L2 (I; (S)−1 ), thus all products on the right-hand side of Eq. 25 are well-defined (Lemma 6) and remain in L2 (I; (S)−1 ).   Fix now an essentially bounded GRP a ∈ W 1,∞ (I ) ⊗ (S)−1,−( p−4) . Then, by Eq. 22 and Lemma 5 it follows that the operator A : W −1,2 (I ) ⊗ (S)−1,− p → W −1,2 (I ) ⊗ (S)−1,− p defined by b  → a♦b is a continuous linear operator. For a fixed c ∈ W01,2 (I ) ⊗ (S)−1,− p consider the operator Fc : W −1,2 (I ) ⊗ (S)−1,− p → R, given by Fc (b ) = a♦b , c. According to Lemma 5 and the Cauchy-Schwartz inequality we have |Fc (b )| = |a♦b , c| ≤ Const b W −1,2 ⊗(S )−1,− p c W 1,2 ⊗(S )−1,− p 0

−1,2

i.e. Fc is a linear continuous operator on the Hilbert space W (I ) ⊗ (S)−1,− p . Due to the Riesz representation theorem there exists a unique fc ∈ W01,2 (I ) ⊗ (S)−1,− p such that Fc (b ) = b , fc .

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This defines a mapping c  → fc , which we will denote by a . In fact, a : c  → fc is the adjoint operator of A. Definition 9 The unique linear continuous mapping a : W01,2 (I ) ⊗ (S)−1,− p → W01,2 (I ) ⊗ (S)−1,− p such that for each b ∈ W −1,2 (I ) ⊗ (S)−1,− p and c ∈ W01,2 (I ) ⊗ (S)−1,− p a♦b , c = b , a (c)

(26)

holds, is called the Wick–adjoint multiplication operator of the generalized random process a ∈ W 1,∞ (I ) ⊗ (S)−1,−( p−4) . In order to solve the Dirichlet problem, we will also consider the Wick–adjoint operator acting on L2 (I ) ⊗ (S)−1 : Let a ∈ L∞ (I ) ⊗ (S)−1,−( p−4) , i.e. a˜ ∈ L∞ (I ) ⊗ (S)−1,− p/2 where a˜ = K p (a) as defined in Eq. 21. Due to Lemma 6 we have that for b ∈ L2 (I ) ⊗ (S)−1,− p , the operator A defined by b  → a♦b maps L2 (I ) ⊗ (S)−1,− p into itself, and it is continuous. Now for c ∈ L2 (I ) ⊗ (S)−1,− p we obtain by Lemma 6 |a♦b , c| ≤ ˜a 2L∞ ⊗(S )−1,− p/2 b L2 ⊗(S )−1,− p c L2 ⊗(S )−1,− p , which leads to the following definition: Definition 10 The unique linear continuous mapping a : L2 (I ) ⊗ (S)−1,− p → L2 (I ) ⊗ (S)−1,− p such that for each b , c ∈ L2 (I ) ⊗ (S)−1,− p a♦b , c = b , a (c)

(27)

holds, is called the Wick–adjoint multiplication operator of the generalized random process a ∈ L∞ (I ) ⊗ (S)−1,−( p−4) . Due to the previous definitions we are able to develop a kind of weak Wick calculus: The Wick-adjoint operator inherits its properties from the classical Wick multiplication—most important, the chain rule holds in weak sense. Lemma 8 For an arbitrary GRP f ∈ W 1,∞ (I ) ⊗ (S)−1,−( p−4) and for arbitrary g ∈ W01,2 (I ) ⊗ (S)−1,− p we have that f  (g) ∈ W01,2 (I ) ⊗ (S)−1,− p and its weak derivative Di ( f  (g)) ∈ L2 (I ) ⊗ (S)−1,− p satisfies:     Di f (g) , v = (Di f ) (g) + f  (Di g), v , (28) for all v ∈ L2 (I ) ⊗ (S)−1,− p and i = 1, 2, . . . , n. Proof From the definition of the Wick–adjoint multiplication we have   Di ( f  (g)), v = − f  (g), Di v = − g, f ♦ Di v = − g, Di ( f ♦v) − (Di f )♦v = Di g, f ♦v + g, (Di f )♦v   = f  (Di g), v + (Di f ) (g) . where we used the chain rule for Wick multiplication (Lemma 7).

 

Later we will see that all properties of the operator L (ellipticity and other assumptions on its coefficients) can be carried over to the formal Wick–adjoint of L.

The Stochastic Weak Maximum Principle

379

4.2 Assumptions on L. The Concepts of Generalized Weak Solutions and Componentwise Solutions The assumptions on L also must be modified to be compatible with the Wick calculus. Thus, we impose additional assumptions on L: There exists λ > 0 such that for all v1 , v2 , . . . vn ∈ W01,2 (I ) ⊗ (S)−1,− p following conditions hold: n n  

aij♦v j, vi  ≥ λ

i=1 j=1

n 

vi 2L2 (I )⊗(S )−1,− p

(ellipticity),

(29)

i=1

aij, b i , ci , d ∈ L∞ (I ) ⊗ (S)−1,−( p−4)

i, j = 1, 2, . . . , n.

(30)

˜ ∈ L∞ (I ) ⊗ (S)−1,− p/2 . Once we have found Recall that Eq. 30 means that a˜ ij, b˜ i , c˜i , d, a solution u ∈ WS of the Dirichlet problem, we will assume that it satisfies the following condition: d♦u, v −

n 

b i ♦u, Di v ≤ 0,

for all v ∈ W01,2 (I ) ⊗ (S)−1,− p , v ≥ 0,

(31)

i=1

which will ensure the uniqueness of this solution. The notation v ≥ 0 for v ∈ 1,2 W01,2 (I ) ⊗ (S)−1,−  p means that all coefficients vα ∈ W0 (I ),α ∈ I , in the chaos expansion v = α∈I vα ⊗ Hα have the property vα ≥ 0. Thus, under conditions (29), (30) and (31) we can prove existence and uniqueness of the generalized weak solution of Eq. 32. Later on we will imply a stronger condition to Eq. 30 in order to prove regularity properties and estimates for Colombeau solutions. Example 3 Let n = 1 and a(x, ω) = 2 + Hε1 (ω), x ∈ I. Then a, a˜ ∈ (L)2 and it satisfies condition (29). Indeed, for any v ∈ W0 S we have a♦v, v = 2 v 2−1,− p + Hε1 ♦v, v−1,− p ≥ 2 v 2−1,− p − |Hε1 ♦v, v−1,− p | " ! p ≥ 2 v 2−1,− p − H˜ ε1 2−1,− p/2 v 2−1,− p = 2 − 2− 2 v 2−1,− p . Example 4 Let aij(x, ω) = exp♦ W(x, ω) = 1+

∞

1 ♦n (x, ω) for all i, j= 1, 2,. . ., n. n=1 n! W  ˜ =1+ ∞ 1W ˜ ♦n ∈ L∞ (I ) ⊗ exp♦ W n=1 n! ♦

Similarly as in Example 2 one can show that (S)−1 i.e. condition (30) is satisfied. Now we will show that exp W satisfies also the ellipticity condition (29). The Dirichlet problem with this particular choice of A = [aij]n×n becomes related to the pressure equation ∇(A♦∇u) = h which was studied in[7].   n  Since i,n j=1 exp♦ W ♦vi , v j = exp♦ W ♦ i=1 vi , nj=1 v j , it suffices to check if 

exp♦ W ♦v, v ≥ λ v 2L2 (I )⊗(S )−1,− p

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S. Pilipovi´c, D. Seleši

for all v ∈ W01,2 (I ) ⊗ (S)−1,− p . Proceeding similarly as in the previous example we obtain exp♦ W ♦v, v ≥ v 2L2 (I )⊗(S )−1,− p − ≥ v 2L2 (I )⊗(S )−1,− p − ≥ v 2L2 (I )⊗(S )−1,− p −

∞  1

W ♦n ♦v L2 (I )⊗(S )−1,− p v L2 (I )⊗(S )−1,− p n! n=1 ∞  Cn n=1

 =

∞  1 |W ♦n ♦v, v| n! n=1

1−

∞  n=1

n!

2 ˜ 2n2

W

L (I )⊗(S )−1,− p/2 v L2 (I )⊗(S )−1,− p

 Cn ˜ 2n

W L2 (I )⊗(S )−1,− p/2 v 2L2 (I )⊗(S )−1,− p . n!

Since the Hermite functions are uniformly bounded (say, by some K), √ ∞ ∞ Cn ˜ 2 ˜ 2n we have

W

K k=1 (2k)− p/2 . Now, n=1 n! W −1,− p/2 ≤ −1,− p/2 ≤ √   ∞ exp C K k=1 (2k)− p/2 − 1 → 0, as p → ∞. Thus, for p large enough,   n ˜ 2n2 1 − ∞ C W

> 0. L (I )⊗(S )−1,− p/2

n=1 n!

We turn now back to our stochastic Dirichlet problem L♦u(x, ω) = h(x, ω) +

n 

Di f i (x, ω),

x ∈ I, ω ∈ ,

i=1

u(x, ω) ∂ I = g(x, ω),

(32)

where the action L♦ onto u ∈ W 1,2 (I ) ⊗ (S)−1,− p is defined in Eq. 24. Applying Definition 4 we obtain L♦u, v =

n 

⎞ ⎛ n n   Di ⎝ aij♦ Dju + b i ♦u⎠ , v + c i ♦ Di u, v + d♦u, v

i=1

=−

j=1

i=1

n  n 

n 

i=1 j=1

i=1

aij♦ Dju, Di v −

b i ♦u, Di v +

n 

c i ♦ Di u, v + d♦u, v.

i=1

for u ∈ W 1,2 (I ) ⊗ (S)−1,− p , v ∈ W01,2 (I ) ⊗ (S)−1,− p . Following [6] we associate a bilinear form B : W 1,2 (I ) ⊗ (S)−1,− p × W01,2 (I ) ⊗ (S)−1,− p → R with L, defined by B(u, v) = −L♦u, v =

n  n  i=1 j=1

aij♦ Dju, Di v +

n  i=1

b i ♦u, Di v−

n  i=1

c i ♦ Di u, v−d♦u, v. (33)

The Stochastic Weak Maximum Principle



381

We call u ∈ W 1,2 (I ) ⊗ (S)−1,− p a generalized weak solution of Eq. 32 if L♦u, v = n n Di f i , v = h, v − i=1  f i , Di v for all v ∈ W01,2 (I ) ⊗ (S)−1,− p , i.e. if h + i=1 B(u, v) = −h, v +

n 

for all v ∈ W01,2 (I ) ⊗ (S)−1,− p .

 f i , Di v

i=1

On the other hand, we will consider also the concept of componentwise solutions: GRP u will be called a componentwise solution of Eq. 32 if L♦u, v =  A n h + i=1 Di f i , v for every test function of the form v(x, ω) = ϕ(x)Hα (ω), where 1,2 ϕ ∈ W0 (I ), and Hα , α ∈ I , are the Fourier-Hermite polynomials.  In other words, if we represent all GRPs by their chaos expansions: h(x, ω) = α∈I hα (x) ⊗ Hα (ω),    ij f i (x, ω) = α∈I fαi (x)⊗ Hα (ω), g(x, ω) = α∈I gα (x)⊗ Hα (ω), aij(x, ω) = α∈I aα (x)⊗   H (ω), b i (x, ω) = α∈I b iα (x) ⊗ Hα (ω), c i (x, ω) = α∈I cαi (x) ⊗ Hα (ω), d(x, ω) = α α∈I dα (x) ⊗ Hα (ω), i, j = 1, 2, . . . , n and search for a solution in form of u(x, ω) = α∈I uα (x) ⊗ Hα (ω), then Eq. 32 obtains the form  γ ∈I

⎛ ⎝

⎞



Lα uβ (x)⎠ ⊗ Hγ (ω) =

α+β=γ



 hγ (x) +

γ ∈I

n 

 Di fγi (x)

⊗ Hγ (ω),

(34)

i=1

 n n i n ij i Di where Lα = i=1 j=1 aα (x)Dj + b α (x) + i=1 cα (x)Di + d(x), α ∈ I , are deterministic linear partial differential operators. This can be reduced to a family of deterministic Dirichlet problems: 

Lα uβ (x) = hγ (x) +

α+β=γ

n 

Di fγi (x),

uγ (x) ∂ I = gγ (x),

γ ∈ I.

i=1

First we calculate uγ for γ = (0, 0, 0, . . .), then uγ for |γ | = 1, and proceed by a recursion method by the length of the multiindex: Once that uβ are known for |β| < |γ |, we find uγ from L(0,0,...) uγ (x) = hγ (x) +

n  i=1

Di fγi (x) −



Lα uβ (x), uγ (x) ∂ I = gγ (x).

(35)

α+β=γ |β|<|γ |

Once we have found their  solutions uγ , γ ∈ I , by regular PDE methods, it remains to prove the series u = γ ∈I uγ ⊗ Hγ converges in W 1,2 (I ) ⊗ (S)−1,− p . This u is then called a componentwise solution. Clearly, if u is a generalized weak solution, then it is also a componentwise solution. Especially, considering the zeroth term, we obtain that the generalized expectation of the generalized weak solution coincides with the weak solution of the deterministic Dirichlet problem, which is obtained by replacing all input data with their generalized expectations. Lemma 9 Let the operator L satisfy condition (30). The bilinear form B(·, ·) given in Eq. 33 is continuous.

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S. Pilipovi´c, D. Seleši

Proof Since the Wick product is distributive with respect to addition, B is indeed bilinear. Continuity follows from Eq. 30, the Cauchy–Schwartz inequality and Lemma 6: |B(u, v)| ≤

n  n n      ij  i  a ♦ Dju + b i ♦u, Di v  +  c ♦ Di u + d♦u, v  i=1 j=1

≤

n n  

i=1

˜aij 2L∞ ⊗(S )−1,− p/2 Dju L2 ⊗(S )−1,− p Di v L2 ⊗(S )−1,− p

i=1 j=1

+

n 

b˜ i 2L∞ ⊗(S )−1,− p/2 u L2 ⊗(S )−1,− p Di v L2 ⊗(S )−1,− p

i=1

+

n 

c˜ i 2L∞ ⊗(S )−1,− p/2 Di u L2 ⊗(S )−1,− p v L2 ⊗(S )−1,− p

i=1

˜ 2∞ + d

L ⊗(S )−1,− p/2 u L2 ⊗(S )−1,− p v L2 ⊗(S )−1,− p ≤C

 n 

Di u L2 ⊗(S )−1,− p

 n 

i=0



Di v L2 ⊗(S )−1,− p

i=0

= C u W 1,2 ⊗(S )−1,− p v W 1,2 ⊗(S )−1,− p 0

  ˜ 2∞ where C = max1≤i, j≤n ˜aij, b˜ i , c˜ i , d

L ⊗(S )−1,− p/2 . Thus, B is indeed a bilinear contin-

uous mapping B : W 1,2 (I ) ⊗ (S)−1,− p × W01,2 (I ) ⊗ (S)−1,− p → R.

 

Now, similarly as in the deterministic case, we can identify the operator L♦ with its unique extension L♦ : W01,2 (I ) ⊗ (S)−1,− p → W −1,2 (I ) ⊗ (S)−1,− p defined through the bilinear form B. Clearly, for fixed u0 ∈ W01,2 (I ) ⊗ (S)−1,− p , the mapping W01,2 (I ) ⊗ (S)−1,− p  v  → B(u0 , v) is linear and continuous, thus defines L♦u0 as an element of W −1,2 (I ) ⊗ (S)−1,− p . Existence of a weak generalized solution is equivalent to surjectivity, while uniqueness is equivalent to injectivity of the mapping L♦. 4.3 The Stochastic Weak Maximum Principle First we introduce the notions of positivity and weak positivity of generalized random processes. Recall that we use two kinds of dual pairings, ·, · and ·, ·W S and we may define weak positivity with respect to either of them. We will consider only the case of the S-antidual pairing ·, ·. •

A GRP v ∈ W01,2 (I ) ⊗ (S)−1,− p is called positive, denoted by v ≥ 0, if it has expansion v(x, ω) = α∈I vα (x) ⊗ Hα (ω), vα ∈ W 1,2 , and vα (x) ≥ 0 for all x ∈ I, α ∈ I.

The Stochastic Weak Maximum Principle

•

• • • •

383

An element u ∈ W −1,2 (I ) ⊗ (S)−1,− p is positive in weak sense, denoted by u ≥ 0 if u, vW S ≥ 0 for all v ∈ W01,2 (I ) ⊗ (S)−1,− p , v ≥ 0. This is equivalent to the fact that if u has expansion u = α∈I uα ⊗ Hα , then for all α ∈ I , uα is positive in weak sense i.e. uα , ϕW ≥ 0 for all ϕ ∈ W01,2 (I ), ϕ ≥ 0. Note that if u ∈ W01,2 (I ) ⊗ (S)−1,− p is positive, then u is also positive in weak sense. This follows from the v ∈ W01,2 (I ) ⊗ (S)−1,− p , v ≥ 0, we have u, v = # fact that for each  − pα ≥ 0. α∈I I uα (x)vα (x)dx(2N) 1,2 Let u ∈ W0 (I ) ⊗ (S)−1,− p . Then u is said to satisfy L♦u ≥ 0 in I in a weak sense, if B(u, v) ≤ 0 for all v ≥ 0. Respectively, L♦u ≤ 0 in a weak sense, if B(u, v) ≥ 0 for all v ≥ 0.  + For u ∈ W 1,2 (I ) ⊗ (S)−1,− p , u(x, ω) = α∈I uα (x) ⊗ Hα (ω),  let u+ be the unique 1,2 + element in W (I ) ⊗ (S)−1,− p given by u (x, ω) = α∈I uα (x) ⊗ Hα (ω) =  α∈I max{uα (x), 0} ⊗ Hα (ω). 1,2 Let u(x, ω) = 0 on ∂ I if u+ (x, ω) ∈ W01,2 (I ) ⊗ (S)−1,− p i.e. if u+ α (x) ∈ W0 (I ) for all α ∈ I .  For u ∈ W 1,2 ⊗ (S)−1,− p , u(x, ω) = α∈I uα (x) ⊗ Hα (ω), uα ∈ W 1,2 (I ), define Supx∈I u(x, ω) to be the element in (S)−1,− p given by  sup uα (x)Hα (ω) Sup u(x, ω) = x∈I

α∈I

=



x∈I

  inf k ∈ R : uα ≤ k Hα (ω)

α∈I

=



kα Hα (ω) = K(ω) ∈ (S)−1,− p .

α∈I

•

•

 2 − pα Clearly, kα ≤ uα W 1,2 , which implies < ∞ and thus α∈I kα (2N) Supx∈I u(x, ω) ∈ (S)−1,− p . Let Inf x∈I u(x, ω) = − Supx∈I (−u(x, ω)). In a similar manner one can define Supx∈∂ I u(x, ω) and Inf x∈∂ I u(x, ω). Note, Inf and Sup are only notations, they do not mean a classical infimum or supremum, since for fixed x ∈ I u(x, ·) is an element of (S)−1,− p which has no partial ordering. Let u ∈ W −1,2 (I ) ⊗ (S)−1,− p and I˜ ⊂ I be an open set. We say that F(x, ω) = 0 in I˜ ×  if F, vW S = 0 for all v ∈ W01,2 (I ) ⊗ (S)−1,− p such that suppv ⊂  I˜ × . Let Spt u(x, ω) =I \ A∈τ A, where τ = { I˜ ⊂ I; u(x, ω) = 0 in I˜ × }. Note that for  u(x, ω) = α∈I uα (x) ⊗ Hα (ω) we have I \ supp uα ∈ τ and thus Spt u(x, ω) ⊆ α∈I supp uα (x).

The next theorem is a weak maximum principle within W 1,2 (I ) ⊗ (S)−1,− p . The proof is inspired by the proof of the classical weak maximum principle in W 1,2 (I ) given in [6], which is now adopted to the setting of chaos expansions. Theorem 2 Assume that the operator L satisfies conditions (29) and (30). (i) Let u ∈ W 1,2 (I ) ⊗ (S)−1,− p satisfy L♦u ≥ 0 and Eq. 31 in I. Then Sup u(x, ω) ≤ Sup u+ (x, ω) x∈I +

x∈∂ I

(i.e. Supx∈∂ I u (x, ω) − Supx∈I u(x, ω) is positive in a weak sense).

384

S. Pilipovi´c, D. Seleši

(ii) Let u ∈ W 1,2 (I ) ⊗ (S)−1,− p satisfy L♦u ≤ 0 and Eq. 31 in I. Then Inf u(x, ω) ≥ Inf u− (x, ω). x∈I

x∈∂ I

Proof (i) Let B(u, v) ≤ 0. Then, using Eq. 31 and boundedness of the coefficients c i we obtain n  n 

aij♦ Dju, Di v ≤

i=1 j=1

n 

c i ♦ Di u, v

i=1

≤C

n 

Di u L2 ⊗(S )−1,− p v L2 ⊗(S )−1,− p

(36)

i=1

for some constant C > 0. If c i = 0, i = 1, 2, . . . , n, then put $ % v(x, ω) = max u(x, ω) − Sup u+ (x, ω), 0 x∈∂ I

=



$ % + max uα (x) − sup uα (x), 0 ⊗ Hα (ω). x∈∂ I

α∈I

Note that v ≥ 0 since vα (x) = max{uα (x) − supx∈∂ I u+ α (x), 0} ≥ 0 for each α ∈ I , and Di v = Di u, i = 1, 2, . . . , n. Now from Eq. 36 and the ellipticity condition (29) we obtain that λ

n 

Di v 2L2 ⊗(S )−1,− p ≤

i=1

n  n   ij a ♦ Djv, Di v ≤ 0, i=1 j=1

i.e. Di v L2 ⊗(S )−1,− p = 0, for all i = 1, 2, . . . , n. Thus, v(x, ω) is constant in the x variable, i.e. v(x, ω) = V(ω) ∈ (S)−1,− p . Now using the Poincaré inequality with respect to the x variable, v L2 ⊗(S )−1,− p ≤ K Dv L2 ⊗(S )−1,− p , we obtain v L2 ⊗(S )−1,− p = 0 i.e. vα (x) = 0 for a.e. x ∈ I and all α ∈ I . Thus, for every α ∈ I we have supx∈I uα (x) − supx∈∂ I u+ α (x) ≤ 0. From this follows that we have also in weak sense Supx∈I u(x, ω) − Supx∈∂ I u+ (x, ω) ≤ 0. Now assume that there exists i ∈ {1, 2, . . . , n} such that c i  = 0. Denote by J a subset of the index set I such that sup u+ α (x) < sup uα (x),

x∈∂ I

α ∈ J.

(37)

x∈I

If J = ∅ the proof is finished, so assume J  = ∅ and let α0 ∈ J . Let k0 ∈ R be such that sup u+ α0 (x) ≤ k0 < sup uα0 (x)

x∈∂ I

x∈I

and v(x, ω) = vα0 (x) ⊗ Hα0 (ω), where vα0 (x) = max{uα0 (x) − kα0 , 0}. Note that vα0 ∈ W01,2 (I ). Let α0 = supp vα0 . We will use the notation · L2 (α0 ) in an obvious way.

The Stochastic Weak Maximum Principle

385

Now we have Di vα0 = Di uα0 for uα0 > kα0 (i.e. for vα0  = 0) and Di vα0 = 0 for uα0 ≤ kα0 (i.e. for vα0 = 0). Now from Eq. 36 and the ellipticity condition we obtain that λ

n 

Di v 2L2 ⊗(S )−1,− p ≤

i=1

n  n n   aij♦ Djv, Di v ≤ C

Di v L2 ⊗(S )−1,− p v L2 ⊗(S )−1,− p , i=1 j=1

i=1 α0

and since Di v L2 ⊗(S )−1,− p = Di vα0 L2 (α0 ) (2N)− p 2 , i = 1, . . . , n, v L2 ⊗(S )−1,− p =

vα0 L2 (α0 ) (2N)

α − p 20

, we obtain n 

Di vα0 L2 (α0 ) ≤

i=1

2C

vα0 L2 (α0 ) . λ

(38)

At this point we will apply the Poincaré inequality 

h L2 (I ) ≤

|I|n(n/2) 2π n/2

n

Dh L2 (I ) ,

for h ∈ W01,2 (I ), where |I| denotes the Lebesgue measure of I. If we denote  = suppDh, then taking Iε ⊃ , where Iε ⊂ I are open sets and Iε → , ε → 0, one obtains 

h L2 () ≤

|| cn

n

Dh L2 () ,

h ∈ W01,2 (I ),

(39)

n/2

2π is the volume of the unit sphere in Rn . where cn = n(n/2) Now by Eqs. 38 and 39 we have



vα0 L2 (α0 ) ≤  ≤

|α0 | cn |α0 | cn

n



Dvα0 L2 (α0 ) ≤

n K

|α0 | cn

n K

n 

Di vα0 L2 (α0 )

i=1

2C

vα0 L2 (α0 ) , λ

for some constant K > 0. Thus,  |α0 | ≥ cn

λ 2KC

 n1

.

Since these inequalities are independent on kα0 , they hold as kα0 → supx∈I uα0 (x). This implies that uα0 attains its supremum over a set of measure strictly greater then zero. From [6, Lemma 7.7] it follows that Duα0 = 0. But this is now in contradiction with the assumption (37), and so supx∈∂ I u+ α0 (x) = supx∈I uα0 (x) must hold. Finally we let α0 run through J , and this proves the theorem.  

386

S. Pilipovi´c, D. Seleši

The uniqueness of the generalized weak solution (and thus also of the componentwise solution) of the homogeneous Dirichlet problem now follows directly from the maximum principle: Corollary 1 Assume that the operator L satisfies Eqs. 29 and 30. Let u ∈ W 1,2 (I ) ⊗ (S)−1,− p satisfy L♦u(x, ω) = 0 and Eq. 31 in I × . Then u = 0.

5 Concluding Remarks •

Theorem 2 and Corollary 1 remain valid also if we replace Eq. 31 by the following condition: d♦u, v +

n  ci ♦u, Di v ≤ 0,

v ∈ W01,2 (I ) ⊗ (S)−1,− p , v ≥ 0.

(40)

i=1

•

This will be used in the second part of this article to prove uniqueness of the Dirichlet problem involving the adjoint operator of L. In the second part of the paper we will consider a more general case involving Colombeau processes. For this purpose we will instead of Eq. 30 imply a stronger condition: Let there exist , ν > 0 such that for all x ∈ I, n 

˜aij(x, ·) 4−1,− p/2 ≤ 2

and

i, j=1 n " 1 1  ! ˜i 2 ˜ ·) 2

b (x, ·) 4−1,− p/2 + c˜ i (x, ·) 4−1,− p/2 + d(x, −1,− p/2 ≤ ν . (41) 2 λ i=1 λ

•

In this paper, and also in its forthcoming second part, we interpret ·, · in Eq. 7 as the scalar product in L2 (I ) ⊗ (S)−1,− p , for a fixed p ∈ N0 . This is needed in order to have both the continuity property of the bilinear form (Lemma 9) and the ellipticity condition (29) hold at the same time. As for defining the notions of derivatives, weak derivatives of GRPS and the notions of weak positivity, one can also interpret ·, · as the usual dual pairing of the dual spaces WS ∗ and W0 S .

Acknowledgements This paper was supported by the project Functional analysis methods, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.

References 1. Adams, R.A.: Sobolev Spaces. Academic, London (1975) 2. Besold, P.: Solutions to stochastic partial differential equations as elements of tensor product spaces. Ph.D. Thesis, Universität Göttingen (2000) 3. Bulla, I.: Dirichlet problem with stochastic coefficients. Stoch. Dyn. 5(4), 555–568 (2005) 4. Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. AMS, Providence (1997) 5. Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4. Academic, New York (1964) 6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1998) 7. Holden, H., Lindstrøm, T., Øksendal, B., Ubøe, J., Zhang, T.: The pressure equation for fluid flow in a stochastic medium. Potential Anal. 4, 655–674 (1995)

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8. Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations—A Modeling. White Noise Functional Approach. Birkhäuser, Boston (1996) 9. Hida, T., Kuo, H.-H., Pothoff, J., Streit, L.: White Noise—an Infinite Dimensional Calculus. Kluwer, Dordrecht (1993) 10. Kuo, H.-H.: White Noise Distribution Theory. CRC, Boca Raton (1996) 11. Pilipovi´c, S., Seleši, D.: Expansion theorems for generalized random processes, wick products and applications to stochastic differential equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(1), 79–110 (2007) 12. Seleši, D.: Algebra of generalized stochastic processes and the stochastic Dirichlet problem. Stoch. Anal. Appl. 26(5), 978–999 (2008) 13. Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, London (1967) 14. Våge, G.: Variational methods for PDEs applied to SPDEs. Math. Scand. 82, 113–137 (1998)