Small ball probabilities for the stochastic heat equation with colored noise

TL;DR

This paper analyzes the small ball probabilities for solutions to a stochastic heat equation driven by colored Gaussian noise, providing estimates under specific conditions on the noise and the coefficient function.

Contribution

It offers new estimates for small ball probabilities of the stochastic heat equation with colored noise, extending previous results to more general noise structures.

Findings

Derived bounds for small ball probabilities of the solution

Extended analysis to colored Gaussian noise in space

Provided conditions on the coefficient function for estimates

Abstract

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