Rate of convergence of uniform transport processes to Brownian sheet

In a previous paper we have constructed a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly in the unit square. Now, a rate of convergence from these processes to Brownian sheet is given.


Introduction
Let W = {W (s, t) : (s, t) ∈ [0, 1] 2 } be a Brownian sheet, i.e. a zero mean real continuous Gaussian process with covariance function E[W (s 1 , t 1 )W (s 2 , t 2 )] = (s 1 ∧ s 2 )(t 1 ∧ t 2 ) for any (s 1 , t 1 ), (s 2 , t 2 ) ∈ [0, 1] 2 . Our aim is to obtain the rate of convergence of strong approximations of the Brownian sheet. This result is not only interesting from a purely mathematical point of view, but are of great interest in order to provide sound approximation strategies to solutions of stochastic partial differential equations whih arise in many fields as physics, biology or finance.
The study of the approximations of the Brownian sheet by uniform transport processes or processes constructed from a Poisson process begins with the proof of the weak convergence. Bardina and Jolis [3] prove that the process where {N (x, y), x ≥ 0, y ≥ 0} is a Poisson process in the plane, converges in law to a Brownian sheet when n goes to infinity and Bardina, Jolis and Rovira [4] extended this result to the d-parameter Wiener processes. On the other hand, Bardina, Ferrante and Rovira [2] constructed a family of processes, starting from a set of independent standard Poisson processes, that has realizations that convergence almost surely to a Brownian sheet. Our purpose is to give the rate of convergence of such approximations. As far as we know, our work is the first rate of convergence for the multiparameter case for this family of approximations..
There exist several literature about strong convergence of uniform transport processes and the study of the corresponding rate of convergence. In the seminal paper of Griego, Heath and Ruiz-Moncayo [11], the authors presented realizations of a sequence of the uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In [9] Gorostiza and Griego extended the result of [1] to the case of diffusions. Again Gorostiza and Griego [10] and Csörgő and Horváth [5] obtained the rate of convergence of the approximation sequence. More recently, Garzón, Gorostiza and León [6] defined a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals, for any Hurst parameter H ∈ (0, 1) and computed the rate of convergence. In [7] and [8] the same authors deal with subfractional Brownian motion and fractional stochastic differential equations. Bardina, Binotto and Rovira [1] proved the strong convergence to a complex Brownian motion and obtained the corresponding rate of convergence.
The structure of the paper is the following. In the next section we recall the approximations that converge almost surely to the Brownian sheet and we present our theorem. In the last section we give the proof of our result. It is based on a combination of the properties of the Brownian sheet and the use of the rate of convergence for the Brownian motion given by Griego, Heath and Ruiz-Moncayo in [11].

Approximations and main result
Let us recall the approximation processes introduced in [2]. For any n and λ > 0, consider the partition of the unit square [0, 1] 2 in disjoint rectangles where ⌊x⌋ denotes the greatest integer less than or equal to x.
} is a Brownian sheet on the unit square, let W k denotes its restriction to each of the above defined rectangles Moreover, puttingW k (t) := n From the paper of Griego, Heath and Ruiz-Moncayo [11] it is known that there exist realizations of uniform transport processes that converge strongly and uniformly on bounded time intervals to Brownian motion. So, we can get an approximation sequence {W (n)k ; n ≥ 1} for each one of the standard Brownian motionsW k ; k ∈ {1, 2, . . . , n λ }. We can state Theorem 1 in [10] for such approximation sequence for any k: of the uniform transport processes on the same probability space as a Brownian motion process a.s., and such that for all q > 0 where α is a positive constant depending on q.
Then, the Brownian sheet is approximated by a process W n such that for any l ∈ {1, 2, . . . , [n λ ]} and t ∈ [0, 1] and {N k , k ≥ 1} is a family of independent standard Poisson processes and {A k , k ≥ 1} is a sequence of independent random variables with law Bernoulli 1 2 , independent of the Poisson processes. Using linear interpolation, define W n (s, t) on the whole unit square as follows: In the following theorem we give our main result, the rate of convergence of these processes: |W n (s, t) − W (s, t)| = 0 a.s. and such that for all β < λ 2 and q > 0 P max where α is a positive constant depending on q.
Notice that the rate of convergence if worse than the rate for the Brownian motion case due to the properties of the Brownian sheet.
The first part of the Theorem has been proved in Theorem 2.1 in [2]. In the next section we give the proof of the rate of convergence.

Proof
We will begin recalling some technical results about submartingales. Let us begin with a version of Theorem 1, page 74 from Imkeller [12].
We will apply this theorem to the process On the other hand, M (1, 1) ψ = exp(B(1, 1)) ψ is a finite constant. Indeed, using that B (1, 1) is a N (0, 1), and this quantity is bigger that 1 for µ sufficiently small and smaller that 1 for µ big enough. We have used that Let us give now the proof of our Theorem: Proof of Theorem 2.2: Our aim is to bound Let us study first P 2 . Using the definitions of W (n)k and W k , for any k, we get Finally, from the rate of convergence from W (n)k andW k , for any k, Theorem 2.1, we get for any q > 0. Let us consider now P 3 . We can write, using the well-known properties of the Brownian sheet |W (r, t)| > K n 3 We have considered that W ( l n λ + r, t) = W (1, t) if l n λ + r > 1. Let us define B(s, t) := n λ 2 W ( s n λ , t), for any (s, t) ∈ [0, 1] 2 . Clearly B is a Brownian sheet and we can bound P 3 as P 3 ≤ (n λ + 1)P max (s,t)∈[0,1] 2 |B(s, t)| > K n 3 n λ 2 .