Weak approximation of the fractional Brownian sheet from random walks ∗

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.


Introduction and main result
Given α, β ∈ (0, 1), a fractional Brownian sheet on R is a two-parameter centered Gaussian process W α, β = {W α, β (t, s), (t, s) ∈ R 2  + } such that For α = β = 1 2 , W α, β coincides with the standard Brownian sheet.It is an extension of fractional Brownian motion B α = {B α t , 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 ∈ ( 1 2 , 1) from random walks in the plane and give an application.
X ε (t, s) with α, β ∈ ( 1 2 , 1) converges in law, as ε tends to zero, to the fractional Brownian sheet W α, β , where {N (x, y), (x, y) ∈ R 2 + } is a standard Poisson process in the plane and the kernel K H given by with H ∈ ( 1 2 , 1) and the normalizing constant c H > 0 given by .
The results of Bardina et 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 where N is a standard Poisson process, converges in law, as ε tends to zero, to the standard Brownian motion W .More works concerning weak approximation for multidimensional parameter process have been studied by many authors (see, for examples, Bardina et 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 if Z ∼ P oiss(λ) then E[(−1) Z ] = exp(−2λ).
Let now {ξ i , i = 1, 2, . ..} be a triangular array of i.i.d.random variables with Eξ (n) i = 0 and E(ξ Sottinen [14] considered the family of processes for n = 1, 2, . .., and showed that the family converges weakly to B H for H ∈ ( 1 2 , 1), where the sequence {K (n) 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.
This paper is organized as follows.In Section 2 we give the proof of Theorem 1.1.
Clearly, when α > 1 2 , β = 1 2 , W α, β is called a fractional noise with Hurst parameter α 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.
Recall that a fractional Brownian sheet admits an integral representation of the form where B is a standard Brownian sheet and K H is the deterministic kernel given by (1.1).
For the deterministic kernel given by (1.1) it is not difficult to see that for all 0 < t 0 < t 0 and 0 < t < t .Let Λ be the group of all mappings λ : [0, T ] × [0, S] → [0, T ] × [0, S] of the form λ(t, s) = (λ 1 (t), λ 2 (s)), where each λ i is continuous, strictly increasing and fixes zero 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 processes Z n are null on the axes, it suffices to prove the following lemma.
Lemma 2.1.Let Z n (t, s) be the family of processes defined by (1.5).Then for any (t, s) < (t , s ), we have 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) < (t , s ), we have Thus, Then, applying the Cauchy-Schwarz inequality, the above term can be bounded by for all 0 < r < r , ν ∈ ( 1 2 , 1) and all n ≥ 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 .
Notice that Eξ (n) = 0 and E 2 ξ (n) = 1, therefore, the above expectation can be computed Using the Cauchy-Schwarz inequality, we get that by Lemma 2.2 and the lemma follows.
Page 5/13 ecp.ejpecp.org Weak approximation of the fractional Brownian sheet 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 processes Z n (t, s) defined by (1.5) converge, as n tends to infinity, to the fractional Brownian sheet in the sense of finite-dimensional distribution.
Proof.For any a 1 , . . ., a d ∈ R and (t 1 , s 1 ), . . ., converges in distribution to a normal random variable with zero mean and variance In fact, the zero mean is trivial.Let us now calculate the limiting variance of Y n .We have By the mean value theorem the above equation is equal to for some s ous and decreasing we get the inner sum in (2.3) is equal to By using the following facts: • The kernel K H with Decompose Y n as follows Now, in order to end the proof we need to obtain the following Lindeberg condition: for all ε > 0. To see that, let us consider the set Noticing that the kernel K H (t, s) with 1 2 < H < 1 is increasing in t and decreasing in s, where A := ( d k=1 a k ) 2 and δ (n) := (2.7) for all i, j = 1, 2, . . ., n, and that Thus, the Lindeberg condition (2.6) holds and the theorem follows.

An application
It is well-known that a fractional Brownian sheet W α, β with β = 1 2 and α > 1 2 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 That is, W α is a Brownian motion in the space variable and a fractional Brownian motion with Hurst parameter α ∈ ( 1 2 , 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 with Dirichlet boundary conditions , where u 0 is a continuous function and W α is the fractional noise with 1 2 < 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 F W t 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 the σ− field of progressively measurable subsets of [0, T ] × Ω.We denote by E the set of step functions on [0, T ] × [0, 1].Let H be the Hilbert space defined as the closure of E with respect to the scalar product According to Nualart-Ouknine [12], the mapping 1 [0,t]×A → W α (t, A) can be extended to an isometry between H and the Gaussian space H 1 (W α ) associated with W α and denoted by where K α is the square integrable kernel given by (1.1).Moreover, for any pair of step functions ϕ and ψ in E we have As a consequence, the operator K * α provides an isometry between the Hilbert space H and L 2 ([0, T ] × [0, 1]).Hence, the Gaussian family {B(t, A), t Denote by , the Green function associated to the heat equation in [0, 1] with Dirichlet boundary conditions.We have Assume that b is bounded, then a P ⊗ B([0, 1])-measurable and continuous random field where the last term is equal to Remark 3.1.We should notice that the mild solution to (3.1), given by (3.2), is understood 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 space C([0, T ] × [0, 1]) we consider the triangular array {ξ    In order to prove Theorem 3.2, we first consider the linear problem, which is amount to establish the convergence in law, in C([0, T ] × [0, 1]), of the solutions of with vanishing initial data and Dirichlet boundary conditions where the solutions of (3.7) and (3.8) are respectively given by (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 Bardina et 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 {X n , n = 1, 2, . ..} is tight, if there exist p, p > 0, δ > 2 and a constant C > 0 such that Lemma 3.4.Let (F, • ) be a normed space and let J, J n , n = 1, 2, . . .be linear maps defined on F with their values in the space L 0 (Ω) of almost surely finite random variables.Assume that there exists a positive constant C such that, for any f ∈ F , and that, for some dense subspace D of F , it holds that J n (f ) converges in law to J(f ), as n tends to infinity, for all f ∈ D.Then, the sequence of random variables {J n (f ), n = 1, 2, . ..} converges in law to J(f ), for any f ∈ F .

x 0 K
(2013), paper 90.Page 8/13 ecp.ejpecp.orgWeak approximation of the fractional Brownian sheet is a space-time white noise, and the process W α has an integral representation of the form W α (t, x) = t 0 α (t, s)B(ds, dy).