Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds

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}Δ_Mu(t,x)+σ(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 $\ddot{I}ƒ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.