<span class="mathjax-formula">$L_p$</span>-theory for the stochastic heat equation with infinite-dimensional fractional noise

DOI:10.1051/ps/2009006 www.esaim-ps.org

L

-THEORY FOR THE STOCHASTIC HEAT EQUATION WITH INFINITE-DIMENSIONAL FRACTIONAL NOISE

Raluca M. Balan

Abstract. In this article, we consider the stochastic heat equation du = (Δu +f(t, x))dt+

_∞

k=1g^k(t, x)δβ^k_t, t ∈ [0, T], with random coefficients f and g^k, driven by a sequence (β^k)_k of i.i.d.

fractional Brownian motions of indexH >1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (β^k)_k, we prove that the equation has a unique solution (in a Banach space of summability exponentp≥2), and this solution is H¨older continuous in both time and space.

Mathematics Subject Classification. 60H15, 60H07.

Received January 15, 2009. Revised April 9, 2009.

1. Introduction

The study of stochastic partial differential equations driven by colored noise has become an active area of research in the recent years, which is viewed as an alternative (with an increased potential for applications) to the classical theory of equations perturbed by space-time white noise (see [5,10,13,26] for fundamental developments – using different approaches – in the white noise case.)

A Gaussian noise is said to be fractional in time, if its temporal covariance structure coincides with that of a fractional Brownian motion (fBm). Recall that a centered Gaussian process (β_t)_t∈[0,T] is a fBm of index H ∈(0,1) if R_H(t, s) := E(β_tβ_s) = (t^2H+s^2H− |t−s|^2H)/2. The caseH >1/2 is referred as the “regular”

case, whereas the case H = 1/2 corresponds to the Brownian motion. (The survey articles [19] and [9] offer more details on the fBm.)

Since the fBm is not a semimartingale, one cannot use the Itˆo calculus, which lies at the foundation of the study of equations driven by white noise. Various methods exist in the literature to circumvent this difficulty, based on the Skorokod integral (e.g.[1,2,4,6,7]), the pathwise generalized Stieltjes integrals (e.g. [21,23,27]), or the “rough paths” analysis (e.g.[15,16]).

The present article is dedicated to the study of the stochastic heat equation with (additive) infinite-dimensional fractional noise:

du(t, x) = (Δu(t, x) +f(t, x))dt+ ∞ k=1

g^k(t, x)δβ^k_t, t∈[0, T], x∈R^d, (1.1)

Keywords and phrases. Fractional Brownian motion, Skorohod integral, maximal inequality, stochastic heat equation.

∗Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

1University of Ottawa, Department of Mathematics and Statistics, 585 King Edward Avenue Ottawa, ON, K1N 6N5, Canada;

rbalan@uottawa.ca, http://aix1.uottawa.ca/~ rbalan

Article published by EDP Sciences c EDP Sciences, SMAI 2011

P

where (β^k)_k is a sequence of i.i.d. fBm’s of index H > 1/2, the solution is defined in the weak sense (using integration against test functions φ ∈ C₀^∞(R^d)), and δβ^k_t is a formal way of indicating that the stochastic integrals (which are used for defining the solution) are interpreted in the Skorohod sense.

Let H_pⁿ(R^d) (n ∈ R, p ≥ 2) be the Sobolev space of all generalized functions on R^d whose derivatives of order k ≤ n lie in L_p(R^d). Our main result shows that for suitable initial condition u₀, and Sobolev-space valued random processes f = {f(t,·)}_t∈[0,T] and g^k = {g^k(t,·)}_t∈[0,T], k ≥ 1, equation (1.1) has a unique H_pⁿ(R^d)-valued solutionu={u(t,·)}_t∈[0,T], andu∈C([0, T], H_pⁿ⁻²(R^d)) a.s., such that

Esup

t≤Tu(t,·)^p_Hn−2

p (R^d)<∞, E _T

0 u(t,·)^p_Hn

p(R^d)dt <∞.

Moreover,ubelongs to the H¨older spaceC^α−1/p([0, T], H_p^n−2β), with probability 1, for any 1/2≥β > α >1/p.

If in addition,γ:=n−2β−d/p >0,uis alsoγ-H¨older continuous in space, sinceH_p^n−2β(R^d)⊂C^γ(R^d). These results provide generalizations to the fractional case of the existing results for the heat equation driven by a sequence (w^k)_k of i.i.d. Brownian motions (see [12,13,22]).

We note that our result cannot be inferred from the results existing in the literature for parabolic equations driven by Hilbert-space valued fractional noise with trace-class covariance operator (e.g.[8,17,25]). Neverthe- less, we should mention the recent related investigations of [21] and [23], using fractional calculus techniques (as opposed to the Malliavin calculus techniques used here), which establish the existence and H¨older continu- ity (in time) of a variational/mild L₂(D)-valued solution for a parabolic initial-boundary value problem with multiplicative fractional noise, whenD⊂R^d is a bounded open set.

Similarly to the Brownian motion case, at the origin of our developments lie two basic tools: (1) a general- ization of the Littlewood-Paley inequality for Banach-space valued functions (Thm.A.2, Appendix); and (2) a suitablep-th moment maximal inequality for the sum of Skorokod integrals with respect to (β^k)_k (Thm. 3.6):

Esup

t≤T

∞ k=1

_t

0 u^k_sδβ_s^k

p

≤ C_p,H,T

⎧⎪

⎪⎨

⎪⎪

⎩ E

_T

0

∞ k=1

|u^k_s|²ds

p/2

+E

_T

0

⎡

⎣ _T

0

_∞

k=1

|D^β_θ^ku^k_s|² _1/(2H)

dθ

⎤

⎦

2H

ds

p/2⎫

⎪⎪

⎬

⎪⎪

⎭

. (1.2)

Compared to the Burkholder-Davis-Gundy inequality (which was used in the Brownian motion case), inequal- ity (1.2) contains an additional term involving the Malliavin derivative D^βku^k of the processu^k with respect to β^k. It is because of this extra term that our developments deviate significantly from the white noise case, and we require that the multiplication coefficientg^k lie in a suitable space of Malliavin differentiable functions with respect to β^k (which in particular, implies thatg^k is measurable with respect toβ^k).

This article is organized as follows. In Section 2, we give some preliminaries on the Malliavin calculus for Hilbert-space valued fractional processes, and we develop a maximal inequality for these processes. In Section 3, we convert the inequality obtained in Section 2 (which speaks about the Skorohod integral with respect to a Hilbert-space valued fractional process), into an inequality which speaks about the sum of Skorohod intregrals with respect to a sequence (β^k)_k of i.i.d. fBm’s. In Section 4, we introduce the stochastic Banach spaces in which we are allowed to select the coefficients f and (g^k)_k. Section 5 is dedicated to the main result, as well as the H¨older continuity of the solution. The appendix contains the generalization of the Littlewood-Paley inequality to Banach space valued functions.

2. Malliavin calculus for fractional processes

In this section, we introduce the basic facts about the Malliavin calculus with respect to (Hilbert-space valued) fractional processes. We refer the reader to [18] and [20] for a comprehensive account on this subject.

Throughout this work, we letH∈(1/2,1) be fixed.

We begin by introducing some Banach spaces and Hilbert spaces of deterministic functions, which are used for the Malliavin calculus with respect to fractional processes.

If V is an arbitrary Banach space, we letEV be the class of all elementary functions φ: [0, T]→V of the form φ(t) =_m

i=11_(ti−1,ti](t)ϕ_i with 0≤t₀< . . . < t_m≤T and ϕ_i∈V. Let|HV|be the space of all strongly measurable functionsφ: [0, T]→V withφ_|HV|<∞, where

φ²_|HV|:=α_{H T}

0

_T

0 φ(t)V φ(s)V|t−s|^2H−2dtds, α_H =H(2H−1).

The spaceEV is dense in|HV|with respect to the norm·_|HV|. It is known that there exists a constantb_H >0 such thatφ_|HV|≤b_HφL_1/H([0,T];V)for anyφ∈L_1/H([0, T];V) (seee.g. relation (11) of [2]).

In particular, ifV =R, we denote EV =E and|HV|=|H|.

We let|H| ⊗ |HV|be the space of all strongly measurable functionsφ: [0, T]²→V with φ_|H|⊗|HV|<∞, where

φ²_|H|⊗|HV|:=α²_H

[0,T]⁴φ(t, θ)V φ(s, η)V |t−s|^2H−2 |θ−η|^2H−2dθdηdsdt.

IfV is a Hilbert space, we letHV be the completion ofEV with respect to the inner product·,·HV defined by:

φ, ψ_HV :=α_{H T}

0

_T

0 φ(t), ψ(s)V|t−s|^2H−2dsdt.

We have:

φ_HV ≤ φ_|HV|≤b_Hφ_{L1/H([0,T];V)}≤b_Hφ_L2([0,T];V), (2.1) and L₂([0, T];V)⊂L_1/H([0, T];V)⊂ |HV| ⊂ HV. In particular, ifV =R, we denote HV =H. The spaceH may contain distributions of order−(2H−1). Note thatHV is isomorphic withH ⊗V, and the inner products in the two spaces are the same.

We let|HV|⊗|HV|be the space of all strongly measurable functionsφ: [0, T]²→V⊗V withφ_|HV|⊗|HV|<

∞, where

φ²_|HV|⊗|HV|:=α²_H

[0,T]⁴φ(t, θ)V⊗V φ(s, η)V⊗V |t−s|^2H−2 |θ−η|^2H−2dθdηdsdt, andH_V ⊗ H_V be the completion ofE_V ⊗ E_V with respect to the inner product·,·_HV⊗HV defined by:

φ, ψHV⊗HV :=α²_H

[0,T]⁴φ(t, θ), ψ(s, η)V⊗V|t−s|^2H−2|θ−η|^2H−2dθdηdsdt.

We have: (seee.g. Lem. 1, [2] for the second inequality below)

φ_HV⊗HV ≤ φ_|HV|⊗|HV|≤b_Hφ_L1/H([0,T]²_;V⊗V)≤b_Hφ_L2([0,T]²_;V⊗V), (2.2) andL₂([0, T]²;V ⊗V)⊂L_1/H([0, T]²;V ⊗V)⊂ |HV| ⊗ |HV| ⊂ HV ⊗ HV.

We begin now to introduce the main ingredients of the Malliavin calculus with respect to fractional processes.

P

Let V be an arbitrary Hilbert space and B = (B(φ))_φ∈HV be a centered Gaussian process, defined on a probability space (Ω,F, P), with covariance:

E(B(φ)B(ψ)) =φ, ψ_HV, ∀φ, ψ∈ H_V. (2.3)

If we letB_t(ϕ) :=B(1_[0,t]ϕ) for anyϕ∈V, t∈[0, T], then

E(B_t(ϕ)B_s(η)) =R_H(t, s)ϕ, ηV, ∀ϕ, η∈V, s, t∈[0, T].

(In particular, ifV =R, thenβ_t:=B(1_[0,t]), t∈[0, T] is a fBm of indexH.) Let

SB:={F =f(B(φ₁), . . . , B(φ_n));f ∈C_b^∞(Rⁿ), φ_i ∈ HV, n≥1}

be the space of all “smooth cylindrical” random variables, where C_b^∞(R^d) denotes the class of all bounded infinitely differentiable functions on Rⁿ, whose partial derivatives are also bounded. Clearly SB ⊂L_p(Ω) for anyp≥1.

TheMalliavin derivativeof an elementF =f(B(φ₁), . . . , B(φ_n))∈ SB, with respect toB, is defined by:

D^BF :=

n i=1

∂f

∂x_i(B(φ₁), . . . , B(φ_n))φ_i.

Note thatD^BF ∈L_p(Ω;HV) for any p≥1; by abuse of notation, we writeD^BF = (D^B_t F)_t∈[0,T] even ifD^B_tF is not a function int. We endowSB with the norm:

F^p_D1,p B

:=E|F|^p+ED^βF^p_HV,

and we letD^1,p_B be the completion ofSB with respect to this norm. The operatorD^B can be extended toD^1,p_B . The adjoint

δ^B : Domδ^B⊂L₂(Ω;HV)→L₂(Ω)

of the operatorD^B, is called theSkorohod integralwith respect toB. The operatorδ^B is uniquely defined by the following relation:

E(F δ^B(U)) =ED^BF, U_H, ∀F ∈D^1,2_B .

Note that E(δ^B(U)) = 0 for any u ∈ Dom δ^B. If U ∈ Dom δ^B, we use the notation U = (U_t)_t∈[0,T] and δ^B(U) =_T

0 U_sδB_s.

IfV is an arbitrary Hilbert space, we let SB(V) :=

⎧⎨

⎩U = m j=1

F_jφ_j;F_j ∈ SB, φ_j∈V, m≥1

⎫⎬

⎭

be the class of all “smooth cylindrical”V-valued random variables. ClearlySB(V)⊂L_p(Ω;V) for anyp≥1.

The Malliavin derivative of an elementU =_m

j=1F_jφ_j ∈ SB(V) is defined byD^BU :=_m

j=1(D^BF_j)φ_j. We haveD^BU ∈L_p(Ω;HV ⊗V) for anyp≥1. We endowSB(V) with the norm:

U^p_D1,p

B (V):=EU^p_V+ED^BU^p_HV⊗V,

and letD^1,p_B (V) be the completion ofSB(V) with respect to this norm. The operator D^B can be extended to D^1,p_B (V).

In particular, ifV =HV, thenD^1,2_β (HV)⊂Dom δ^B. If U ∈D^1,2_B (HV) then D^BU ∈L₂(Ω;HV ⊗ HV); by abuse of notation, we writeD^BU = (D^B_t U_s)_s,t∈[0,T].

The space D^1,2_B (HV) is viewed as a “suitable” class of Skorohod integrands with respect to B. For any U ∈D^1,2_B (HV), we have:

E|δ^B(U)|² = EU²_HV +E

D^BU,(D^BU)^∗_HV⊗HV)

≤ EU²_HV +ED^BU²_HV⊗HV =U²_D1,2

B (HV), (2.4)

where (D^BU)^∗ is the adjoint ofD^BU inHV ⊗ HV.

The following result is a consequence of Meyer’s inequalities.

Proposition 2.1 (Prop. 2.4.4 of [18]). Let p > 1 and U ∈ D^1,p_B (HV). Then U lies in the domain of δ^B in L_p(Ω) and

E|δ^B(U)|^p ≤C_H,p

E(U)^p_HV +ED^BU^p_HV⊗HV , whereC_H,p is a constant depending onH andp.

As a consequence of Proposition2.1, (2.1) and (2.2), we obtain:

E|δ^B(U)|^p≤C_H,pb_H{E(U)^p_{L1/H([0,T];V)}+ED^BU^p_L1/H([0,T]2;V⊗V)}. (2.5) We denote by D^1,p_B (|HV|) the set of all elementsU ∈D^1,p_B (HV), such thatU ∈ |HV| a.s.,D^BU ∈ |HV| ⊗ |HV| a.s., andU_D^1,p

B (|HV|)<∞, where U^p_D1,p

B (|HV|) := EU^p_|HV|+ED^BU^p_|HV|⊗|HV|.

The following result generalizes Theorem 4 of [2] to the case ofV-valued fractional processes.

Theorem 2.2. Let 1/2< H <1,p >1/H and0< ε < H−1/p. Then, there exists a constant C depending on H, p, εandT such that

Esup

t≤T

_t

0 U_sδB_s

^p ≤ C

⎧⎨

⎩ T

0 E(Us)^1/(H−ε)_V ds

_p(H−ε)

+E

⎡

⎣ _T

0

_T

0 D^B_θU_s^1/H_V⊗Vdθ _H−ε^H

ds

⎤

⎦

p(H−ε)⎫

⎪⎬

⎪⎭ (2.6)

for any processU = (U_t)_t∈[0,T]∈D^1,p_B (|HV|)for which the right-hand side of (2.6) is finite.

Proof. The argument is similar to the one used in the proof of Theorem 4 of [2]. We include it for the sake of completeness. Let α= 1−1/p−ε.

By writing_t

0U_sδB_s=c_αt

0(t−r)^−α_r

0 U_s(r−s)^α−1δB_s

dr, and using H¨older’s inequality, we obtain:

Esup

t≤T

_t

0 U_sδB_s

^p≤c_α,pE _T

0

_r

0 U_s(r−s)^α−1δB_s ^pdr,

P

wherec_α,pis a constant depending onαandp. Using (2.5), we have:

Esup

t≤T

_t

0 U_sδB_s

^p ≤ c_α,p,H T

0

_r

0 E(Us)^1/H_V (r−s)^(α−1)/Hds _pH

dr

+E _T

0

_r

0

_T

0 D_θ^BU_s^1/H_V⊗V(r−s)^(α−1)/Hdθds _pH

dr

⎫⎬

⎭,

where c_α,p,H is a constant which depends on α, p andH. The result follows by applying Hardy-Littlewood

inequality (p. 119 of [24]).

Whenp≥2, the previous theorem leads to the following result.

Corollary 2.3. Let 1/2< H <1 and p≥2 be arbitrary. Then, there exists a constant C depending on H, p andT such that

Esup

t≤T

_t

0

U_sδB_s

^p ≤ C

⎧⎨

⎩ _T

0 E(U_s)²_Vds _p/2

+E

⎡

⎣ _T

0

_T

0 D^B_θU_s^1/H_V⊗Vdθ _2H

ds

⎤

⎦

p/2⎫

⎪⎬

⎪⎭ (2.7)

for any processU = (U_t)_t∈[0,T]∈D^1,p_B (|HV|)for which the right-hand side of (2.7) is finite.

Proof. The result follows by applying Theorem2.2withε < H−1/2 and using the fact thatφL1/(H−ε)([0,T])≤

C_Tφ_L2([0,T]) for anyφ∈L₂([0, T]).

3. The Maximal inequality

The goal of this section is to translate the p-th moment maximal inequality given by Corollary 2.3 into a similar inequality (in the l₂-norm) for a sequence (u^k)_k of Skorohod integrable processes, with respect to a sequence (β^k)_k of i.i.d. fBm’s. The idea is to recover a Gaussian process B (as in Sect. 2) from (β^k)_k, and to construct a Skorohod integrable process U (with respect to B) from the sequence (u^k)_k, such that δ^B(U1_[0,t]) =_∞

k=1δ^βk(u^k1_[0,t]) for allt∈[0, T] a.s.

Let β^k = (β_t^k)_t∈[0,T], k ≥ 1 be a sequence of i.i.d. fBm’s of Hurst index H > 1/2, defined on the same probability space (Ω,F, P). LetV be an arbitrary Hilbert space, and (e_k)_k a complete orthonormal system in V.

The first result shows that it is possible to construct a centered Gaussian process B with covariance (2.3), from the sequence (β^k)_k. This result is probably well-known; we state it for the sake of completeness.

Lemma 3.1. Let (φ^k)_k ⊂ Hbe such that_∞

k=1φ^k2_H<∞. Then:

a)ϕ^(N):=_N

k=1φ^ke_k ∈ HV for allN ≥1, and there existsϕ:=_∞

k=1φ^ke_k ∈ HV such thatϕ^(N)−ϕHV → 0 asN → ∞. We have:

ϕ²_HV = ∞ k=1

φ^k2_H; (3.1)

b) B^(N)(ϕ) :=_N

k=1β^k(φ^k)∈L₂(Ω) for anyN ≥1, and there existsB(ϕ) :=_∞

k=1β^k(φ^k)∈L₂(Ω) such that E|B^(N)(ϕ)−B(ϕ)|² → 0 as N → ∞. The process B = {B(ϕ)}ϕ∈HV is Gaussian with mean zero and

covariance (2.3). In particular, for any t∈[0, T], ϕ∈V, we have:

B_t(ϕ) :=B(1_[0,t]ϕ) = ∞ k=1

ϕ, ekVβ_t^k in L₂(Ω). (3.2)

Proof. a) The sequence{ϕ^(N)}N is Cauchy inHV, since (ϕ^(N)−ϕ^(M))(t) =_N

k=M+1φ^k(t)e_kfor anyN > M≥ 1, and hence

ϕ^(N)−ϕ^(M)2_HV = α_H N k=M+1

_T

0

_T

0 φ^k(t)φ^k(s)|t−s|^2H−2dsdt

= N k=M+1

φ^k2_H→0, asM, N → ∞.

In particular,ϕ^(N)2_HV =_N

k=1φ^k2_H. By letting N → ∞, we obtain (3.1).

b) The sequence {B^(N)(ϕ)}N is Cauchy in L₂(Ω), since B^(N)(ϕ)−B^(M)(ϕ) = _N

k=M+1β^k(φ^k) for any N > M ≥1, and hence

E|B^(N)(ϕ)−B^(M)(ϕ)|²= N k=M+1

E|β^k(φ^k)|²= N k=M+1

φ^k2_H→0, asM, N → ∞.

To prove (3.2), note that 1_[0,t]ϕ = _∞

k=1φ^ke_k, where φ^k = 1_[0,t]ϕ, ekV. It follows that B(1_[0,t]ϕ) = _∞

k=1β^k(φ^k) =_∞

k=1ϕ, e_kVβ_t^k.

We begin now to explore the relationship between the Malliavin derivatives with respect to (β^k)_k and the Malliavin derivative with respect toB.

An immediate consequence of (3.2) is thatβ^k_t =B(1_[0,t]e_k) for anyt∈[0, T], and hence

β^k(φ) =B(φe_k), ∀φ∈ H. (3.3)

Let F = f(β^k(φ₁), . . . , β^k(φ_n))∈ S_β^k be arbitrary, with f ∈ C_b^∞(Rⁿ) and φ_i ∈ H. Then ϕ_i :=φ_ie_k ∈ HV, F =f(B(ϕ₁), . . . , B(ϕ_n))∈ SB, and

D_t^BF = n i=1

∂f

∂x_i(B(ϕ₁), . . . , B(ϕ_n))ϕ_i = _n

i=1

∂f

∂x_i(β^k(φ₁), . . . , β^k(φ_n))φ_i

e_k

= (D_t^βkF)e_k.

From here we conclude thatS_β^k⊂ SB, and for anyF ∈ S_β^k, D^BFHV =D^βkFH, F_D^1,p

B =F_D^1,p

βk

, ∀p≥1.

It follows thatD^1,p_βk ⊂D^1,p_B for anyp≥1, and

D^BF = (D^βkF)e_k, for anyF ∈D^1,2_βk. (3.4)

P

Ifu=_m

j=1F_jφ_j ∈ S_β^k(H) is arbitrary, withF_j∈ S_β^k andφ_j ∈ H, then ue_k ∈D^1,2_B (HV) and D^B(ue_k) =

m j=1

(D^BF_j)φ_je_k = m j=1

(D^βkF_j)φ_je_k⊗e_k= (D^βku)e_k⊗e_k.

In general, ifu∈D^1,2_βk(H), thenue_k ∈D^1,2_B (HV) and

D^B(ue_k) = (D^βku)e_k⊗e_k. (3.5)

Moreover, we have the following result:

Lemma 3.2. If u^k ∈ D^1,2_βk(H), then _N

k=1u^ke_k ∈ D^1,2_B (HV), D^B(_N

k=1u^ke_k) = _N

k=1(D^βku^k)e_k ⊗e_k, and _N

k=1u^ke_k²_D1,2

B (HV)=_N

k=1u^k2_D1,2 βk(H).

Proof. The result follows from the definitions of the norms inD^1,2_B (HV), respectivelyD^1,2_βk(H), and the following two identities:

N k=1

u^ke_k

2 HV

= α_{H T}

0

_T

0

_N

k=1

u^k_te_k, N l=1

u^l_se_l

V

|t−s|^2H−2dsdt

= α_H N k,l=1

_T

0

_T

0 u^k_tu^l_sek, e_lV|t−s|^2H−2dsdt

= N k=1

u^k2_H

D^B _N

k=1

u^ke_k

2 HV⊗HV

= α²_H

[0,T]⁴

_N

k=1

D_θ^B(u^k_te_k), N l=1

D^B_η(u^l_se_l)

V⊗V

×|t−s|^2H−2|θ−η|^2H−2dθdηdsdt

= α²_H N k,l=1

[0,T]⁴(D^β_θ^ku^k_t)(D^β_η^ku^l_s)ek⊗e_k, e_l⊗e_lV⊗V

×|t−s|^2H−2|θ−η|^2H−2dθdηdsdt

= N k=1

D^βku^k2_H⊗H,

where we used (3.5) for the second-last equality above.

We need an auxiliary result.

Lemma 3.3. LetX be a normed space andy_N, x_N,n, x_n, x∈X be such that: lim_n→∞sup_N≥1yN−x_N,n= 0, lim_N→∞xN,n−x_n= 0for all n, andlim_n→∞xn−x= 0. Thenlim_N→∞yN −x= 0.

Proof. We useyN −x ≤ yN−x_N,n+xN,n−x_n+xn−x.

The previous observations allow us to extend Lemma 3.1to the case of random integrands.

Theorem 3.4. Let u^k∈D^1,2_βk(H)for allk≥1, such that ∞

k=1

u^k2_D1,2

βk(H)<∞. (3.6)

Then:

a) U^(N):=_N

k=1u^ke_k ∈D^1,2_B (HV) for anyN ≥1, and there exists U :=_∞

k=1u^ke_k ∈D^1,2_B (HV)such that U^(N)−U_D^1,2

B (HV)→0 asN → ∞. We have: D^BU =_∞

k=1(D^βku^k)e_k⊗e_k and U²_D1,2

B (HV)= ∞ k=1

u^k2_D1,2

βk(H). (3.7)

b) the sequence W^(N):=_N

k=1δ^βk(u^k), N≥1 has a limit in L₂(Ω), which coincides with δ^B(U). We write δ^B(U) =

∞ k=1

δ^βk(u^k) in L₂(Ω). (3.8)

Proof. a) By Lemma3.2,{U^(N)}N is a Cauchy sequence inD^1,2_B (HV), since:

U^(N)−U^(M)2_D1,2 B (HV)=

N k=M+1

u^k2_D1,2

βk(H)→0, asM, N → ∞.

Hence, U := lim_N→∞U^(N) exists in D^1,2_B (HV), and D^BU = lim_N→∞D^BU^(N) in L₂(Ω;HV ⊗ HV). Also, U^(N)2_D1,2

B (HV)=_N

k=1u^k2_D1,2

βk(H), and relation (3.7) follows by lettingN → ∞. b) By inequality (2.4) (applied forV =RandB =β^k), we have:

N k=M+1

E|δ^βk(u^k)|²≤ ^N

k=M+1

u^k2_D1,2

βk(H)→0, asM, N → ∞,

i.e. the sequence {W^(N)}N is Cauchy inL₂(Ω). We letW be the limit of{W^(N)}N in L₂(Ω). We now prove that W =δ^B(U) (inL₂(Ω)).

Step 1. Suppose thatu^k ∈ S_β^k(H) for allk,i.e. u^k =_mk

j=1F_j^kφ^k_j for some F_j^k ∈ S_β^k and φ^k_j ∈ H. Since U^(N)→U inD^1,2_B (HV),δ^B(U^(N))→δ^B(U) inL₂(Ω). On the other hand_N

k=1δ^βk(u^k)→W inL₂(Ω). Hence, it suffices to prove that:

δ^B(U^(N)) = N k=1

δ^βk(u^k). (3.9)

Note thatU^(N)=_N

k=1

_mk

k=1F_j^kφ^k_je_k∈ SB(HV), sinceF_j^k ∈ S_β^k⊂S_B andφ^k_je_k∈ HV. Relation (3.9) follows from (3.3) and (3.4), since:

δ^B(U^(N)) = N k=1

mk

j=1

F_j^kB(φ^k_je_k)− N k=1

mk

j=1

D^BF_j^k, φ^k_je_kHV

N k=1

δ^βk(u^k) = N k=1

mk

j=1

F_j^kβ^k(φ^k_j)− N k=1

mk

j=1

D^βkF_j^k, φ^k_jH. (We used relation (1.9) of [18], for the equalities above.)

Step 2. Suppose that u^k ∈ D^1,2_βk(H) for all k. For any ε > 0, there exists u^k_ε ∈ S_β^k(H) such that u^k_ε −u^k_D^1,2

βk(H) < ε/2^k; hence _∞

k=1u^k_ε²_D1,2

βk(H) < ∞. By part a), U_ε := _∞

k=1u^k_εe_k ∈ D^1,2_B (HV) and