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.

This version strengthens the main result by proving it under more general assumptions. The previous version is recovered as a special case. Journal reference added