Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds

TL;DR

This paper investigates the probability that the solution to a stochastic heat equation on a compact manifold remains close to zero, providing estimates for small deviations under Gaussian noise with specific regularity conditions.

Contribution

It offers new small ball probability estimates for the stochastic heat equation on compact manifolds with Gaussian noise, extending understanding of solution behavior in this geometric setting.

Findings

Derived explicit small ball probability bounds

Extended results to manifold settings with colored spatial noise

Provided conditions under which estimates hold

Abstract

We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}\Delta_Mu(t,x)+\sigma(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\dot{W}$ is a centered Gaussian noise that 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 $u(0,x)\equiv 0$.