Weak convergence for the stochastic heat equation driven by Gaussian white noise

TL;DR

This paper establishes the weak convergence of solutions to a stochastic heat equation driven by Gaussian white noise, using approximations of the noise and analyzing the convergence of the associated laws in a continuous function space.

Contribution

It provides sufficient conditions for the convergence in law of approximate solutions of the SPDE driven by noise families to the true white noise driven solution, focusing on the linear problem.

Findings

Convergence in law of approximate solutions to the SPDE driven by noise families.

Tightness of the laws of the solutions was established.

Identification of the limit law through finite-dimensional distribution convergence.

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 $n\in \mathbb{N}$ such that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued {\it mild} solution of the SPDE perturbed by this family of noises converges in law, in the space $\mathcal{C}([0,T]\times [0,1])$ 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 $n \int_0^t (-1)^{N(n^2 s)} ds$, 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.