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

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.