Small ball probabilities for the stochastic heat equation with colored noise

Abstract

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