On weak convergence of stochastic heat equation with colored noise

Abstract

In this work we are going to show weak convergence of a probability measure corresponding to the solution of the following nonlinear stochastic heat equation $\frac{\partial}{\partial t} u_{t}(x) = \frac{\kappa}{2} \Delta u_{ t}(x) + \sigma(u_{t}(x))\eta_\alpha$ with colored noise $\eta_\alpha$ to the measure corresponding to the solution of the same equation but with white noise $\eta$ as $\alpha \uparrow 1$ on the space of continuous functions with compact support. The noise $\eta_\alpha$ is assumed to be colored in space and its covariance is given by $\operatorname{E} \left[ \eta_\alpha(t,x) \eta_\alpha(s,y) \right] = \delta(t-s) f_\alpha(x-y)$ where $f_\alpha$ is the Riesz kernel $f_\alpha(x) \propto 1/\left|x\right|^\alpha$. We will also state a result about continuity of measure in $\alpha$, for $\alpha \in (0,1)$. We will work with the classical notion of weak convergence of measures.