Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise

Abstract

In this paper, we consider a quasi-linear stochastic heat equation on [0, 1], with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the realvalued mild solution of the SPDE perturbed by this family of noises converges in law, in the space ([0, T ] × [0, 1]) of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock, which states that the family of processes n t 0 (−1) N (n 2 s) ds, where N is a standard Poisson process, converges in law to a Brownian motion. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

“…The first requirement is that θ n ∈ L 2 ([0, 1] 2 ) a.s., and then there are the following conditions (see hypotheses 1.1, 1. Note that the latter expression is almost equal to that in the right hand-side of equation (31) in the proof of [3,Prop. 4.4].…”

“…This makes that both I n 1 and I n 2 can be bounded by the sum of two terms of the form I n j,1 + I n j,2 , j = 1, 2, respectively, where I n j,1 involves f (t 1 , x 1 ) and I n j,2 involves f (t 2 , x 2 ). Then, once all cosinus are simply bounded by 1, one observe that the resulting four terms are completely analogous as those appearing in the proof of Lemma 4.2 in [3], and can be treated using the same kind of arguments. Thus, we obtain that I n 1,1 ≤ Plugging everything together and using (32), we conclude the proof.…”

“…Owing to [3,Sec. 3], equation (31) admits a unique solution U i n whose paths are continuous almost surely.…”

“…We refer to Section 5 for the precise statement of the above-mentioned conditions. In [3], the authors apply their main result to two important families of random fields that approximate the Brownian sheet: the Donsker kernels in the plane and the Kac-Stroock processes, where the latter are defined by θ n (t, x) := n √ tx (−1) N ( processes in [3] (see Section 4 therein), and also on some technical estimates contained in the proof of the tightness result given in Proposition 3.1 of the present paper. Eventually, we note that the kind of convergence results that are obtained in the present paper assure that the limit processes, which in our case correspond to the complex Brownian sheet and the solution to the stochastic heat equation, are robust when used as models in practical situations.…”

“…The above lemma allows us to prove the following proposition. In fact, its proof follows exactly the same lines as that of Proposition 4.1 in [3] and therefore will be omitted.…”