Small ball probabilities for the stochastic heat equation with colored noise

Jiaming Chen

Published 2022-02-09Version 1

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