Small ball probabilities for the stochastic heat equation with colored noise

Abstract

We consider the stochastic heat equation on the 1-dimensional torus $T≔\left(-1,1\right)$ with periodic boundary conditions: ${\partial }_{t}u(t,x)={\partial }_x^{2}u(t,x)+\sigma (t,x,u)\dot{F}(t,x),x\in T,t\in R_{+},$where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time but colored in space. Assuming that $\sigma$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.