Small ball probabilities for the fractional stochastic heat equation driven by a colored noise

Jiaming Chen

Published 2022-07-27Version 1

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