On weak convergence of stochastic wave equation with colored noise on R

In this paper, we study the following stochastic wave equation on the real line ∂ₜ² uα=∂ₓ² uα+b(u_α)+σ(u_α)ηα. The noise η_α is white in time and colored in space with covariance structure E[η_α(t,x)η_α(s,y)]=δ(t-s)f_α(x-y) where f_α(x)∝ |x|-α is the Riesz kernel. We state the continuity of the solution u_α in terms of α for α∈ (0,1) with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also prove that the probability measure induced by u_α converges weakly to the one induced by the solution in the space-time white noise case as α goes to $1$.