Convergence in law for quasi-linear SPDEs

TL;DR

This paper establishes conditions under which solutions to quasi-linear stochastic wave and heat equations driven by Gaussian noise converge in law, extending previous results to more general noise spectral measures including fractional and Riesz kernels.

Contribution

It provides new sufficient conditions for convergence in law of solutions to quasi-linear SPDEs with general spectral measures, broadening the scope of existing convergence results.

Findings

Convergence in law of solutions under general spectral measures.

Application to anisotropic fractional noise and other specific cases.

Extension of existing results to more general noise types.

Abstract

We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $\mu_n$. We allow the Fourier transform of $\mu_n$ to be a genuine distribution. Let $u^n$ be the mild solution to these equations. We provide sufficient conditions on the measures $\mu_n$ and the initial data to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $\mu$, where $\mu_n\to\mu$ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $d=1$.