On weak convergence of stochastic wave equation with colored noise on $\mathbb{R}$

TL;DR

This paper investigates the weak convergence of solutions to a stochastic wave equation driven by colored noise on the real line, establishing conditions under which the probability measures of solutions depend continuously on the noise's spatial covariance parameter.

Contribution

It proves the continuity of the solution's probability measure with respect to the spatial covariance parameter in the noise, extending understanding of stochastic wave equations with colored noise.

Findings

Proved weak convergence of solutions as the noise covariance varies.

Established conditions for the continuity of the solution's law in the space of continuous functions.

Provided examples illustrating the applicability of the main theorem.

Abstract

In this paper, we study the following stochastic wave equation on the real line $\partial_t^2 u_{\alpha}=\partial_x^2 u_{\alpha}+b\left(u_\alpha\right)+\sigma\left(u_\alpha\right)\eta_{\alpha}$. The noise $\eta_\alpha$ is white in time and colored in space with a covariance structure $\mathbb{E}[\eta_\alpha(t,x)\eta_\alpha(s,y)]=\delta(t-s)f_\alpha(x-y)$ where $f_\alpha$ is continuous with respect to $\alpha$ in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $u_\alpha$, in terms of $\alpha$, with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of $f_\alpha$ such that our theorem applies to.