On weak convergence of stochastic heat equation with colored noise

TL;DR

This paper proves that solutions to a nonlinear stochastic heat equation with spatially colored noise converge weakly to solutions with white noise as the noise's color parameter approaches 1, demonstrating measure continuity.

Contribution

It establishes weak convergence of the solution measures for the stochastic heat equation with colored noise to the white noise case as the color parameter tends to 1, including measure continuity results.

Findings

Weak convergence of measures as $ ext{color parameter} o 1$

Continuity of measure in the parameter $ ext{alpha} ext{ for } ext{alpha} ext{ in } (0,1)

Convergence on the space of continuous functions with compact support

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.