On weak convergence of stochastic heat equation with colored noise

Abstract

In this work we are going to show weak convergence of probability measures. The measure corresponding to the solution of the following one dimensional nonlinear stochastic heat equation ∂∂tut(x)=κ2∂2∂x2ut(x)+σ(ut(x))ηα with colored noise ηα will converge to the measure corresponding to the solution of the same equation but with white noise η, as α↑1. Function σ is taken to be Lipschitz and the Gaussian noise ηα is assumed to be colored in space and its covariance is given by E[ηα(t,x)ηα(s,y)]=δ(t−s)fα(x−y) where fα is the Riesz kernel fα(x)∝1/|x|α. We will work with the classical notion of weak convergence of measures, that is convergence of probability measures on a space of continuous function with compact domain and sup–norm topology. We will also state a result about continuity of measures in α, for α∈(0,1).