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: $${\partial }_{t}u(t,x)={\partial }_x^{2}u(t,x)+\ddot{I}ƒ(t,x,u)\dot{F}(t,x),x\in \mathbb{T},t\in {\mathbb{R}}_{+},$$ where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time and colored in space. Assuming that $\ddot{I}ƒ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.

Published in Stochastic Processes and their Applications