Weak approximation of the fractional Brownian sheet from random walks

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ISSN:1083-589X in PROBABILITY

Weak approximation of the fractional Brownian sheet from random walks

^∗

Zhi Wang

^†

Litan Yan

^‡

Xianye Yu

^§

Abstract

In this paper, we show an approximation in law of the fractional Brownian sheet by random walks. As an application, we consider a quasilinear stochastic heat equation with Dirichlet boundary conditions driven by an additive fractional noise.

Keywords: Fractional Brownian sheet; random walks; stochastic heat equation; weak convergence.

AMS MSC 2010:60B10; 60G15; 60H15.

Submitted to ECP on June 17, 2013, final version accepted on November 11, 2013.

1 Introduction and main result

Given α, β ∈ (0,1), a fractional Brownian sheet on Ris a two-parameter centered Gaussian process

W^{α, β} ={W^{α, β}(t, s),(t, s)∈R²+}

such that E

W^{α, β}(t, s)W^{α, β}(t⁰, s⁰)

=1 2

t^2α+t^02α− |t⁰−t|^2α

·1 2

s^2β+s^02β− |s⁰−s|^2β . Forα=β = ¹₂, W^{α, β} coincides with the standard Brownian sheet. It is an extension of fractional Brownian motionB^α={B_t^α, t ≥0} to two-parameter case. In this paper, we will be interested in the weak approximation of the fractional Brownian sheet with α, β∈(¹₂,1)from random walks in the plane and give an application.

Recently, Bardinaet al. [6] (see also Tudor [16] for a similar approximation in the Besov space) proved that the family of processes

X_ε(t, s) := 1 ε²

Z 1 0

K_α(t, u)K_β(s, v)√

uv(−1)^N⁽^u^ε^,^v^ε⁾dudv

∗Supported by the NSFC (11171062), Innovation Program of Shanghai Municipal Education Commission (12ZZ063) and the Research Project of Education of Zhejiang Province (Y201326507)

†Department of Mathematics, College of Science, Donghua University, P.R. China, and School of Science, Ningbo University of Technology, P.R. China

‡Department of Mathematics, College of Science, Donghua University, P.R. China E-mail:[email protected](corresponding author)

§Department of Mathematics, College of Science, Donghua University, P.R. China

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withα, β ∈(¹₂,1)converges in law, asεtends to zero, to the fractional Brownian sheet W^{α, β}, where{N(x, y),(x, y)∈R²+} is a standard Poisson process in the plane and the kernelKH given by

K_H(t, s) = (H−1

2)c_Hs¹²^−H Z t

s

u^H−¹²(u−s)^H−³²du (1.1) withH ∈(¹₂,1)and the normalizing constantcH >0given by

c_H=

s 2HΓ(³₂−H) Γ(H+¹₂)Γ(2−2H).

The results of Bardinaet al. [6] and Tudor [16] have been inspired by the following relationship between the standard one-parameter Poisson process and the standard Brownian motion proved by Stroock [15]: the family of processes

yε(t) :=1 ε

Z t 0

(−1)^N⁽^s^ε⁾ds,

whereN is a standard Poisson process, converges in law, as ε tends to zero, to the standard Brownian motionW. More works concerning weak approximation for multidimensional parameter process have been studied by many authors (see, for examples, Bardinaet al. [3, 5, 6]). In these references, the methods for obtaining the corresponding approximation sequences are Poisson processes due to their good properties such as independent increments and that ifZ ∼P oiss(λ)thenE[(−1)^Z] = exp(−2λ).

Let now {ξ_i⁽ⁿ⁾, i = 1,2, . . .} be a triangular array of i.i.d. random variables with Eξ_i⁽ⁿ⁾= 0andE(ξ_i⁽ⁿ⁾)²= 1. Then the sequence of stochastic processes

W_t⁽ⁿ⁾:= 1

√n

bntc

X

i=1

ξ_i⁽ⁿ⁾, t∈[0, T], n= 1,2, . . .

converges weakly to a standard Brownian motionW, wherebxcdenotes the greatest integer not exceedingx. According to the next integral representation of the fractional Brownian motionB^H with Hurst indexH ∈(¹₂,1):

B_t^H= Z t

0

KH(t, s)dWs, t≥0, (1.2)

Sottinen [14] considered the family of processes{Z⁽ⁿ⁾}

Z_t⁽ⁿ⁾:=

Z t 0

K_H⁽ⁿ⁾(t, s)dW_s⁽ⁿ⁾=

bntc

X

i=1

n Z _nⁱ

i−1 n

K_H(bntc

n , s)ds 1

√nξ⁽ⁿ⁾_i , t∈[0, T]

forn = 1,2, . . ., and showed that the family converges weakly to B^H forH ∈ (¹₂,1), where the sequence{K_H⁽ⁿ⁾(t,·), n= 1,2, . . .}is an approximation toKH(t,·)defined by

K_H⁽ⁿ⁾(t, s) :=n Z s

s−_n¹

KH(bntc

n , u)du, n= 1,2, . . . . (1.3) Motivated by this, in the present paper we consider the approximation of fractional Brownian sheet by random walks in the plane, and our main result is to explain and prove the following theorem.

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Theorem 1.1. Letα > ¹₂, β > ¹₂ and letn

ξ_i,j⁽ⁿ⁾, i, j= 1,2, . . .o

be an independent family of identically distributed and centered random variables withE(ξ_i,j⁽ⁿ⁾)² = 1. Forn ≥1, (t, s)∈[0, T]×[0, S], we set

Bn(t, s) := 1 n

bntc

X

i=1 bnsc

X

j=1

ξ_i,j⁽ⁿ⁾ (1.4)

and

Zn(t, s) : = Z t

0

Z s 0

K_α⁽ⁿ⁾(t, v)K_β⁽ⁿ⁾(s, u)Bn(dv, du)

=n

bntc

X

i=1 bnsc

X

j=1

ξ_i,j⁽ⁿ⁾ Z _nⁱ

i−1 n

Z _n^j

j−1 n

Kα(bntc

n , v)Kβ(bnsc

n , u)dudv.

(1.5)

where the kernelK_·is given by (1.1)and the sequence{K·⁽ⁿ⁾, n= 1,2, . . .} of approximation to K· defined by (1.3). Then, {Zn} converges weakly in the Skorohod space D([0, T]×[0, S]) to the fractional Brownian sheetW^{α, β} in the plane.

This paper is organized as follows. In Section 2 we give the proof of Theorem 1.1.

Clearly, whenα > ¹₂, β = ¹₂,W^{α, β} is called a fractional noise with Hurst parameterα which is introduced in Nualart-Ouknine [12]. Thus, as an application of Theorem 1.1, in Section 3 we consider the approximation solution of a one-dimensional quasi-linear stochastic heat equation driven by fractional noise.

2 Proof of the Theorem 1.1

To prove Theorem 1.1, we first recall some facts. For a deeper discussion we refer the reader to see Ayacheet al. [1], Cairoli-Walsh [8], Decreusefond-Üstünel [9], Kamont [10].

Let(Ω,F, P)be a complete probability space and let{Ft,s; (t, s)∈[0, T]×[0, S]}be a family of sub-σ-fields ofF ^{such that}

(C1) Ft,s⊆Ft⁰,s⁰ for anyt≤t⁰, s≤s⁰; (C2) F0,0contains all null sets ofF^;

(C3) for each z ∈[0, T]×[0, S],Fz =∩z<z⁰Fz⁰, wherez = (t, s)< z⁰ = (t⁰, s⁰)denotes the partial order on[0, T]×[0, S], meaning thatt < t⁰ands < s⁰.

Given(t, s)<(t⁰, s⁰), we denote by∆t,sX(t⁰, s⁰)the increment of the processX over the rectangle((t, s),(t⁰, s⁰)], that is,∆_t,sX(t⁰, s⁰) =X(t⁰, s⁰)−X(t, s⁰)−X(t⁰, s) +X(t, s).

Recall that a fractional Brownian sheet admits an integral representation of the form W^{α, β}(t, s) =

Z t 0

Z s 0

Kα(t, v)Kβ(s, u)B(dv, du), (t, s)∈[0, T]×[0, S], (2.1) whereBis a standard Brownian sheet andKHis the deterministic kernel given by (1.1).

For the deterministic kernel given by (1.1) it is not difficult to see that Z t⁰₀

t₀

(K_H(t⁰, x)−K_H(t, x))²dx≤C_H(t⁰₀−t₀)^2−2H for all0< t0< t⁰₀and0< t < t⁰.

Let Λ be the group of all mappings λ : [0, T]×[0, S] → [0, T]×[0, S] of the form λ(t, s) = (λ1(t), λ2(s)), where eachλi is continuous, strictly increasing and fixes zero

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and one. Denote byD=D([0, T]×[0, S])the Skorohod space of functions on[0, T]×[0, S]

are continuous from above with limits from below and equipDwith the metric d(x, y) := inf{min(kx−yλk,kλk) :λ∈Λ},

wherekx−yλk= sup{|x(t, s)−y(λ(t, s))|: (t, s)∈[0, T]×[0, S]} andkλk= sup{|λ(t, s)− (t, s)| : (t, s)∈[0, T]×[0, S]}. Under this metric,D is a separable and complete metric space.

Now, we can prove Theorem 1.1, and we split the proof in several results. We first prove the tightness. Using the criterion given by Bickel-Wichura [7], and notice that our processesZnare null on the axes, it suffices to prove the following lemma.

Lemma 2.1. Let Zn(t, s) be the family of processes defined by (1.5). Then for any (t, s)<(t⁰, s⁰), we have

sup

n

E[(∆t,sZn(t⁰, s⁰))⁴]≤16^α+β(t⁰−t)^4α(s⁰−s)^4β.

In order to prove Lemma 2.1 we need the next technical result.

Lemma 2.2. Let Z_n(t, s) be the family of processes defined by (1.5). Then for any (t, s)<(t⁰, s⁰), we have

E[(∆_t,sZ_n(t⁰, s⁰))²]≤4^α+β(t⁰−t)^2α(s⁰−s)^2β.

Proof. First, we observe that

∆_t,sZ_n(t⁰, s⁰) = Z t⁰

t

Z s⁰ s

K_α⁽ⁿ⁾(bnt⁰c

n , v)−K_α⁽ⁿ⁾(bntc

n , v) K_β⁽ⁿ⁾(bns⁰c n , u)

−K_β⁽ⁿ⁾(bnsc n , u)

Bn(dv, du)

=

bnt⁰c

X

i=1 bns⁰c

X

j=1

n Z _nⁱ

i−1 n

Z _n^j

j−1 n

K_α⁽ⁿ⁾(bnt⁰c

n , v)−K_α⁽ⁿ⁾(bntc n , v)

·

K_β⁽ⁿ⁾(bns⁰c

n , u)−K_β⁽ⁿ⁾(bnsc n , u)

dudvξ_i,j⁽ⁿ⁾.

Thus,

E[∆t,sZn(t⁰, s⁰)]²=

bnt⁰c

X

i=1

(√ n

Z _nⁱ

i−1 n

(Kα(bnt⁰c

n , v)−Kα(bntc

n , v))dv)²

·.

bns⁰c

X

j=1

(√ n

Z _n^j

j−1 n

(K_β(bns⁰c

n , u)−K_β(bnsc

n , u))du)².

Then, applying the Cauchy-Schwarz inequality, the above term can be bounded by

bnt⁰c

X

i=1

Z _nⁱ

i−1 n

(K_α(bnt⁰c

n , v)−K_α(bntc n , v))²dv

bns⁰c

X

j=1

Z _n^j

j−1 n

(K_β(bns⁰c

n , u)−K_β(bnsc

n , u))²du

≤ Z t⁰

0

(Kα(bnt⁰c

n , v)−Kα(bntc n , v))²dv

Z s⁰ 0

(Kβ(bns⁰c

n , u)−Kβ(bnsc

n , u))²du

=

bnt⁰c − bntc n

2α

bns⁰c − bnsc n

2β

.

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Let now 0 < r < r⁰ and ¹₂ < ν < 1. We then see that nr⁰ −nr ≥ 1 implies that

bnr⁰c−bnrc n

2ν

≤ |2(r⁰−r)|^2ν. Conversely,nr⁰−nr <1implies that eitherr⁰andrbelong to a same subinterval [^m_n,^m+1_n ) for some integer m, and hence

bnr⁰c−bnrc n

2ν

= 0. It follows that

bnr⁰c − bnrc n

2ν

≤ |2(r⁰−r)|^2ν

for all0< r < r⁰,ν∈(¹₂,1)and alln≥1. This completes the proof.

We are now ready to prove Lemma 2.1.

Proof of Lemma 2.1. First, we observe that we can write

E[∆t,sZn(t⁰, s⁰)]⁴=E ^bnt

0c

X

i=1 bns⁰c

X

j=1

n Z _nⁱ

i−1 n

Z _n^j

j−1 n

(K_α⁽ⁿ⁾(bnt⁰c

n , v)−K_α⁽ⁿ⁾(bntc n , v))

·(K_β⁽ⁿ⁾(bns⁰c

n , u)−K_β⁽ⁿ⁾(bnsc

n , u))dudvξ_i,j⁽ⁿ⁾ 4

.

Notice thatEξ⁽ⁿ⁾= 0andE²ξ⁽ⁿ⁾= 1, therefore, the above expectation can be computed as

bnt⁰c

X

i=1 bns⁰c

X

j=1 bnt⁰c

X

k=1 bns⁰c

X

l=1

(√ n

Z _n^j

j−1 n

(Kβ(bns⁰c

n , u)−Kβ(bnsc

n , u))du)²

·(√ n

Z _nⁱ

i−1 n

(K_α(bnt⁰c

n , v)−K_α(bntc

n , v))dv)²

·(√ n

Z _n^l

l−1 n

(K_β(bns⁰c

n , u)−K_β(bnsc

n , u))du)²

·(√ n

Z _n^k

k−1 n

(K_α(bnt⁰c

n , v)−K_α(bntc

n , v))dv)²

=





bnt⁰c

X

i=1

(√ n

Z _nⁱ

i−1 n

(K_α(bnt⁰c

n , v)−K_α(bntc

n , v))dv)²





2

·





bns⁰c

X

j=1

(√ n

Z ^j_n

j−1 n

(K_β(bns⁰c

n , u)−K_β(bnsc

n , u))du)²





2

.

Using the Cauchy-Schwarz inequality, we get that

E[∆t,sZn(t⁰, s⁰)]⁴≤





bnt⁰c

X

i=1

Z _nⁱ

i−1 n

(Kα(bnt⁰c

n , v)−Kα(bntc n , v))²dv





2

·





bns⁰c

X

j=1

Z _n^j

j−1 n

(K_β(bns⁰c

n , u)−K_β(bnsc

n , u))²du





2

≤

bnt⁰c − bntc n

4α

bns⁰c − bnsc n

4β

≤16^α+β(t⁰−t)^4α(s⁰−s)^4β by Lemma 2.2 and the lemma follows.

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Now, it suffices to show that the law of all possible weak limits is the law of a fractional Brownian sheet.

Theorem 2.3. The family of processesZn(t, s)defined by (1.5) converge, asntends to infinity, to the fractional Brownian sheet in the sense of finite-dimensional distribution.

Proof. For anya1, . . . , ad∈R^and(t1, s1), . . . ,(td, sd)∈[0, T]×[0, S]. We claim that

Yn:=

d

X

k=1

akZn(tk, sk)

converges in distribution to a normal random variable with zero mean and variance

E

d

X

k=1

akW^{α, β}(tk, sk)

!²

. (2.2)

In fact, the zero mean is trivial. Let us now calculate the limiting variance ofY_n. We have

(σ⁽ⁿ⁾)²:=E(Yn)²=

d

X

k,l=1

akaln²

bnTc

X

i=1 bnSc

X

j=1

Z _nⁱ

i−1 n

Z _n^j

j−1 n

Kα(bntkc

n , v)Kβ(bnskc

n , u)dudv

· Z _nⁱ

i−1 n

Z _n^j

j−1 n

Kα(bntlc

n , v)Kβ(bnslc

n , u)dudv

=

d

X

k,l=1

a_ka_l

bnTc

X

i=1

n Z _nⁱ

i−1 n

K_α(bnt_kc n , v)dv

Z _nⁱ

i−1 n

K_α(bnt_lc n , v)dv

·

bnSc

X

j=1

n Z _n^j

j−1 n

K_β(bnskc n , u)du

Z ^j_n

j−1 n

K_α(bnslc n , u)du.

By the mean value theorem the above equation is equal to

d

X

k,l=1

akal

1 n

bnTc

X

i=1

Kα(bntkc

n , s⁽ⁿ⁾_i,k)Kα(bntlc n , s⁽ⁿ⁾_i,l )1

n

bnSc

X

j=1

Kβ(bnskc

n , s⁽ⁿ⁾_j,k)Kβ(bnslc n , s⁽ⁿ⁾_j,l)

(2.3) for somes⁽ⁿ⁾_i,k, s⁽ⁿ⁾_i,l ∈(ⁱ⁻¹_n ,_nⁱ]ands⁽ⁿ⁾_j,k, s⁽ⁿ⁾_j,l ∈(^j−1_n ,_n^j]. Since the kernelK_·(t,·)is continuous and decreasing we get the inner sum in (2.3) is equal to

1 n

bnTc

X

i=1

Kα(bntkc

n , v⁽ⁿ⁾_i )Kα(bntlc n , v_i⁽ⁿ⁾)1

n

bnSc

X

j=1

Kβ(bnskc

n , u⁽ⁿ⁾_j )Kβ(bnslc

n , u⁽ⁿ⁾_j ) (2.4) for some

v_i⁽ⁿ⁾∈h

min(s⁽ⁿ⁾_i,k, s⁽ⁿ⁾_i,l )i

⊆ i−1

n , i n

; u⁽ⁿ⁾_j ∈h

min(s⁽ⁿ⁾_j,k, s⁽ⁿ⁾_j,l)i

⊆ j−1

n , j n

.

By using the following facts:

• The kernelKH with ¹₂ < H <1is continuous with respect to both arguments;

• The mapst7→ ^bntc_n ,s7→ ^bnsc_n converge uniformly to the identity map in[0, T]×[0, S],

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we see that (2.4) is a Riemann type sum. It follows that (2.3) converges to

d

X

k,l=1

a_ka_l Z T

0

K_α(t_k, s)K_α(t_l, s)ds Z S

0

K_β(s_k, s)K_β(s_l, s)ds=E(

d

X

k=1

a_kW^{α, β}(t_k, s_k))².

DecomposeYnas follows Y_n=

bnTc

X

i=1 bnSc

X

j=1

nξ⁽ⁿ⁾_i,j

d

X

k=1

a_k Z _nⁱ

i−1 n

K_α(btkc n , v)dv

Z _n^j

j−1 n

K_β(bskc n , u)du

:=

bnTc

X

i=1 bnSc

X

j=1

Y_i,j⁽ⁿ⁾.

(2.5)

Now, in order to end the proof we need to obtain the following Lindeberg condition:

n→∞lim 1 (σ⁽ⁿ⁾)²

bnTc

X

i=1 bnSc

X

j=1

Eh

(Y_i,j⁽ⁿ⁾)²1

{|Y_i,j⁽ⁿ⁾|>εσ⁽ⁿ⁾}

i

= 0 (2.6)

for allε >0. To see that, let us consider the set n|Y_i,j⁽ⁿ⁾|> εo

=n

(Y_i,j⁽ⁿ⁾)²> ε²o .

Noticing that the kernelK_H(t, s)with ¹₂ < H <1is increasing intand decreasing ins, we get

(Y_i,j⁽ⁿ⁾)²=n²(ξ_i,j⁽ⁿ⁾)²

d

X

k=1

a_k Z _nⁱ

i−1 n

K_α(bt_kc n , v)dv

Z _n^j

j−1 n

K_β(bs_kc n , u)du

!²

≤n²(ξ_i,j⁽ⁿ⁾)²A Z _nⁱ

i−1 n

K_α(T, v)dv Z _n^j

j−1 n

K_β(S, u)du

!2

≤(ξ_i,j⁽ⁿ⁾)²A Z _nⁱ

i−1 n

K_α²(T, v)dv Z _n^j

j−1 n

K_β²(S, u)du

≤(ξ_i,j⁽ⁿ⁾)²A Z _n¹

0

K_α²(T, v)dv Z _n¹

0

K_β²(S, u)du= (ξ_i,j⁽ⁿ⁾)²Aδ⁽ⁿ⁾,

whereA:= (Pd

k=1a_k)²andδ⁽ⁿ⁾:=R¹_n

0 K_α²(T, v)dvR_n¹

0 K_β²(S, u)du, which deduces n|Y_i,j⁽ⁿ⁾|> εσ⁽ⁿ⁾o

⊆n

(ξ_i,j⁽ⁿ⁾)²Aδ⁽ⁿ⁾> ε²(σ⁽ⁿ⁾)²o

. (2.7)

It follows that Eh

(Y_i,j⁽ⁿ⁾)²1

{|Y_i,j⁽ⁿ⁾|>εσ⁽ⁿ⁾}

i≤Eh

(ξ_i,j⁽ⁿ⁾)²Aδ⁽ⁿ⁾1

{(ξ_i,j⁽ⁿ⁾)²Aδ⁽ⁿ⁾>ε²(σ⁽ⁿ⁾)²}

i

for alli, j= 1,2, . . . , n, and that 1

(σ⁽ⁿ⁾)²

bnTc

X

i=1 bnSc

X

j=1

Eh

(Y_i,j⁽ⁿ⁾)²1

{|Y_i,j⁽ⁿ⁾|>εσ⁽ⁿ⁾}

i

≤ 1

(σ⁽ⁿ⁾)²

bnTc

X

i=1 bnSc

X

j=1

Eh

(ξ_i,j⁽ⁿ⁾)²Aδ⁽ⁿ⁾1_{(ξ(n)

i,j)²Aδ⁽ⁿ⁾>ε²(σ⁽ⁿ⁾)²}

i

≤Eh (ξ⁽ⁿ⁾)²1

{(ξ⁽ⁿ⁾_i,j)²Aδ⁽ⁿ⁾>ε²(σ⁽ⁿ⁾)²}

i→0 (n→ ∞)

becauseδ⁽ⁿ⁾ →0. Thus, the Lindeberg condition (2.6) holds and the theorem follows.

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3 An application

It is well-known that a fractional Brownian sheet W^{α, β} with β = ¹₂ and α > ¹₂ is called the fractional noise with Hurst parameter α, denoted by W^α, which is first introduced in Nualart-Ouknine [12]. Obviously, it is a zero mean Gaussian process with the covariance function

E[W^α(t, x)W^α(s, y)] = 1 2

t^2α+s^2α− |t−s|^2α

(x∧y).

That is,W^αis a Brownian motion in the space variable and a fractional Brownian motion with Hurst parameterα∈(¹₂, 1)in the time variable.

In the sequel, as an application to Theorem 1.1 we consider the approximation solution (in law) of the stochastic heat equation

∂U

∂t −∂²U

∂x² =b(U) +∂²W^α

∂t∂x , (3.1)

with Dirichlet boundary conditions

U(t,0) =U(t,1) = 0, t∈[0, T]

and initial conditionU(0, x) =u0(x), x∈[0,1], whereu0is a continuous function andW^α is the fractional noise with ¹₂ < α <1. This is a one-dimensional quasi-linear stochastic heat equation on[0,1]which was first studied by Nualart-Ouknine [12].

For each t ∈ [0, T], let Ft^W be the σ− field generated by the random variables {W^α(t, A), t ∈[0, T], A ∈ B[0,1]}and the sets of probability zero,P be theσ−field of progressively measurable subsets of[0, T]×Ω. We denote byE the set of step functions on[0, T]×[0,1]. LetHbe the Hilbert space defined as the closure ofE with respect to the scalar product

h1_[0,t]×A, 1_[0,s]×BiH=E[W^α(t, A)W^α(s, B)].

According to Nualart-Ouknine [12], the mapping1_[0,t]×A→W^α(t, A)can be extended to an isometry between H and the Gaussian space H1(W^α) associated with W^α and denoted by

ϕ7→W^α(ϕ) :=

Z

[0,t]×A

ϕ(s, y)W^α(ds, dy).

Consider the linear operatorK_α^∗fromE ^toL²([0, T])defined by K_α^∗(ϕ) =K_α(T, s)ϕ(s, x) +

Z T s

(ϕ(r, x)−ϕ(s, x))∂K_α

∂r (r, s)dr,

whereK_αis the square integrable kernel given by (1.1). Moreover, for any pair of step functionsϕandψinE ^{we have}

hK_α^∗(ϕ), K_α^∗(ψ)i_L2([0,T]×[0,1])=hϕ, ψiH, because

(K_α^∗1_[0,t]×A)(s, x) =K_α(t, s)1_[0,t]×A(s, x).

As a consequence, the operatorK_α^∗ provides an isometry between the Hilbert spaceH andL²([0, T]×[0,1]). Hence, the Gaussian family {B(t, A), t∈[0, T], A∈ B[0,1]} defined by

B(t, A) =W^α((K_α^∗)⁻¹(1_[0,t]×A)),

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is a space-time white noise, and the processW^αhas an integral representation of the form

W^α(t, x) = Z t

0

Z x 0

Kα(t, s)B(ds, dy).

Denote by Gt(x, y) = 1

√2πt

∞

X

n=−∞

e⁻^{(y−x−2n)2}^4t +e⁻^(y+x−2n)2^4t

= 2

∞

X

n=1

sin(nπx)sin(nπy)e⁻ⁿ²^π²^t,

(t, x, y)∈[0, T]×[0,1]², the Green function associated to the heat equation in[0,1]with Dirichlet boundary conditions. We have

0≤Gt(x, y)≤ 1

√2πte^−(y−x)2^4t , t >0, (x, y)∈[0,1]².

Assume thatbis bounded, then aP ⊗ B([0,1])-measurable and continuous random field U ={U(t, x),(t, x)∈[0, T]×[0,1]}is a solution to (3.1) if and only if

U(t, x) = Z 1

0

G_t(x, y)u₀(y)dy+ Z t

0

Z 1 0

G_t−s(x, y)b(U(s, y))dyds

+ Z t

0

Z 1 0

G_t−s(x, y)W^α(ds, dy),

(3.2)

where the last term is equal to

W^α(1[0,t](·)Gt−·(x,·)) = Z t

0

Z 1 0

K_α^∗Gt−s(x, y)B(ds, dy).

It follows from Nualart-Ouknine [12] that (3.1) admits a unique solution satisfying (3.2), providedα∈(¹₂,1)andbis Lipschitz function with the linear growth.

Remark 3.1. We should notice that the mild solution to (3.1), given by (3.2), is under- stood in the generalized sense defined by Walsh [17] in the case of a space-time white noise.

To study the approximation solution of (3.1) in the spaceC([0, T]×[0,1])we consider the triangular array {ξ_i⁽ⁿ⁾, i = 1,2, . . .} of i.i.d. random variables withEξ_i⁽ⁿ⁾ = 0 and E(ξ_i⁽ⁿ⁾)²= 1, as in Theorem 1.1, and define the processes

Un(t, x) = Z 1

0

Gt(x, y)u0(y)dy+ Z t

0

Z 1 0

Gt−s(x, y)b(Un(s, y))dyds

+ Z t

0

Z 1 0

K_α^∗Gt−s(x, y)θn(s, y)dyds, n= 1,2, . . .

(3.3)

whereθn(t, x),(t, x)∈[0, T]×[0,1]stands for the Donsker kernel given by θ_n(t, x) =n

∞

X

i=1

∞

X

j=1

ξ_i,j⁽ⁿ⁾1_[i−1

n ,_nⁱ)×[^j−1_n ,_n^j)(t, x). (3.4) Observe that since θn have square integrable paths, the integrals in (3.3) are well defined. Standard arguments yield existence and uniqueness of solution for (3.3).

Notice that Z t

0

Z x 0

θ_n(s, y)dyds= 1 n

bntc

X

i=1 bnxc

X

j=1

ξ⁽ⁿ⁾_i,j =B_n(t, x), n= 1,2, . . . . (3.5)

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We see, as an application of Theorem 1.1, that W_n^α(t, x) :=

Z t 0

Z x 0

Kα(t, s)Bn(ds, dy) = Z t

0

Z x 0

Kα(t, s)θn(s, y)dyds, (3.6) converges in law to fractional noiseW^α. Our main object of this section is to explain and prove the following theorem.

Theorem 3.2. Let{θn(t, x),(t, x) ∈ [0, T]×[0,1]}, n = 1,2, . . . be the Donsker kernel given in (3.4). Assume thatu0 : [0,1]→ Ris a continuous function and b : R → R^is Lipschitz. Then, the family {Un, n = 1,2, . . .} defined by (3.3)converges in law, as n tends to infinity, in the spaceC([0, T]×[0,1]), to the mild solutionU of (3.1), given by (3.2).

In order to prove Theorem 3.2, we first consider the linear problem, which is amount to establish the convergence in law, inC([0, T]×[0,1]), of the solutions of

∂Xn

∂t −∂²Xn

∂x² = ∂²W_n^α

∂t∂x , (3.7)

with vanishing initial data and Dirichlet boundary conditionsU(t,0) =U(t,1) = 0, t∈ [0, T], towards the solution of

∂X

∂t −∂²X

∂x² = ∂²W^α

∂t∂x , (3.8)

where the solutions of (3.7) and (3.8) are respectively given by Xn(t, x) =

Z t 0

Z 1 0

K_α^∗Gt−s(x, y)θn(s, y)dyds (3.9) and

X(t, x) = Z t

0

Z 1 0

K_α^∗G_t−s(x, y)B(ds, dy). (3.10) We will make use of the following results, which is a quotation of Theorem 2.1 and Lemma 2.2 in Mellali-Ouknine [11] (see, also Theorem 2.2 and Lemma 2.3 in Bardinaet al.[4]).

Lemma 3.3. Let {X_n, n = 1,2, . . .} be a family of random variables taking values in C([0, T]×[0,1]). The family of the laws of {Xn, n = 1,2, . . .} is tight, if there exist p, p⁰ >0, δ >2and a constantC >0such that

sup

n≥1

E|Xn(0,0)|^p⁰ <∞

and

sup

n≥1

E|Xn(t⁰, x⁰)−Xn(t, x)|^p< C(|x⁰−x|+|t⁰−t|)^δ

for allt, t⁰∈[0, T], x, x⁰ ∈[0,1].

Lemma 3.4. Let(F,k·k)be a normed space and letJ, J_n, n= 1,2, . . .be linear maps defined onFwith their values in the spaceL⁰(Ω)of almost surely finite random variables.

Assume that there exists a positive constantC such that, for anyf ∈F, sup

n≥1

E|Jn(f)| ≤Ckfk, E|J(f)| ≤Ckfk,

and that, for some dense subspace D of F, it holds that Jn(f) converges in law to J(f), as n tends to infinity, for all f ∈ D. Then, the sequence of random variables {Jn(f), n= 1,2, . . .}converges in law toJ(f), for anyf ∈F.

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We also will use the following lemmas which are given in Bardinaet al.[4] and Bally et al.[2].

Lemma 3.5. Let{θn(t, x),(t, x)∈[0, T]×[0,1]},n= 1,2, . . .be the Donsker kernels and letm≥10be some even number. Then, there exists a positive constantC_msuch that

E Z T

0

Z 1 0

f(t, x)θn(t, x)dxdt

!m

≤Cm

Z T 0

Z 1 0

f²(t, x)dxdt

!^m₂

, (3.11)

for alln≥1and allf ∈L²([0, T]×[0,1]).

Lemma 3.6. (i) Letα∈(³₂,3). Then, for allt∈[0, T]andx, y∈[0,1], Z t

0

Z 1 0

|G_t−s(x, z)−G_t−s(y, z)|^αdzds≤C|x−y|^3−α.

(ii) Letα∈(1,3). Then, for alls, t∈[0, T]such thats≤tandx∈[0,1], Z s

0

Z 1 0

|G_t−r(x, y)−G_s−r(x, y)|^αdydr≤C|t−s|^3−α² .

(iii) Under the same hypothesis as (ii), we have Z t

s

Z 1 0

|Gt−r(x, y)|^αdydr≤C|t−s|^3−α² .

Proposition 3.7. The family{Xn, n= 1,2, . . .}given by (3.9)is tight inC([0, T]×[0,1]). Proof. First, we observe that

X_n(t⁰, x⁰)−X_n(t, x) = Z t

0

Z 1 0

K_α^∗(G_t⁰_−s(x⁰, y)−G_t⁰_−s(x, y))θ_n(s, y)dyds +

Z t 0

Z 1 0

K_α^∗(G_t⁰_−s(x, y)−G_t−s(x, y))θ_n(s, y)dyds

+ Z t⁰

t

Z 1 0

K_α^∗G_t⁰_−s(x⁰, y)θ_n(s, y)dyds

≡I1+I2+I3

for allt < t⁰. It follows from Lemma 3.5 that sup

n≥1

E|Xn(t⁰, x⁰)−Xn(t, x)|^m≤Cm

Z t 0

Z 1 0

[K_α^∗(Gt⁰−s(x⁰, y)−Gt⁰−s(x, y))]²dyds ^m2

+Cm

Z t 0

Z 1 0

[K_α^∗(Gt⁰−s(x, y)−Gt−s(x, y))]²dyds ^m2

+Cm

Z t 0

Z 1 0

(K_α^∗Gt⁰−s(x⁰, y))²dyds ^m2

≡Cm(J1+J2+J3).

Using the continuous embedding established in Pipiras-Taqqu [13]

L^α¹([0, T]×[0,1])⊂ H,

and the inequality in Lemma 3.6, we obtain J₁≤C_α

Z t 0

Z 1 0

(G_t⁰_−s(x⁰, y)−G_t⁰_−s(x, y)^α¹dyds ^mα

≤C_α|x⁰−x|^(3α−1)m,

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and similarly, we also haveJ₂, J₃ ≤C_α|t⁰−t|^(3α−1)^m². Consequently, we have sup

n≥1

E|X_n(t⁰, x⁰)−X_n(t, x)|^m≤C_α,m[|t⁰−t|^(3α−1)^m² +|x⁰−x|^(3α−1)m],

and the proposition follows from Lemma 3.3.

Proposition 3.8. The family{X_n, n= 1,2, . . .}defined by (3.9) converges to the process X given by (3.10), in the sense of finite-dimensional distributions, asntends to infinity, in the spaceC([0, T]×[0,1]).

Proof. Let us consider the normed space(F:=L²([0, T]×[0,1]),k · k2). Set Jⁿ(f) :=

Z t 0

Z 1 0

f(s, y)θ_n(s, y)dyds, J(f) :=

Z t 0

Z 1 0

f(s, y)B(ds, dy),

where

f(s, y) =

m

X

j=1

aj1_[0,s_j_](s)K_α^∗Gs_j−s(yj, y).

Then,JⁿandJ define two linear applications onF. By Lemma 3.5 and the continuous embedding as above, we obtain

sup

n≥1

E|Jⁿ(f)| ≤Ckfk₂.

Notice that from the computations of the proof of ( Nualart-Ouknine [12], Lemma 5), and applying Lemma 3.6 we obtain

Z t 0

Z 1 0

(K_α^∗Gt−r(x, y))²dydr≤Cα( Z t

0

Z 1 0

(Gt−s(x, y))^α¹dydr)^2α≤Cαt^3α−1<∞,

which implies

E|J(f)| ≤Ckfk2. Combining this with Lemma 3.4, we complete the proof.

As a consequence of the above two propositions, we can see that the family{Xn, n≥ 1}defined by (3.9) converges in law to the Gaussian processX defined by (3.10).

Finally, in a similar way as Theorem 4.5 in Mellali-Ouknine [11] we can obtain the next theorem, and Theorem 3.2 follows as a direct consequence.

Theorem 3.9. Let{θ_n(t, x),(t, x) ∈ [0, T]×[0,1]}, n = 1,2, . . . be the Donsker kernel given in (3.4). Assume that u0 : [0,1] → R is a continuous function and b : R → R is Lipschitz. If the family {Xn, n ≥ 1} defined by (3.9) converges in law, as n tends to infinity, to the Gaussian processX defined by (3.10). Then, the family {Un, n∈N} defined by (3.3)converges in law, asntends to infinity, in the spaceC([0, T]×[0,1]), to the mild solutionU of (3.1).

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Acknowledgments. The authors would like to express their sincere gratitude to the associate editor and the anonymous referees for their valuable comments.

Weak approximation of the fractional Brownian sheet from random walks