Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise

TL;DR

This paper investigates the probability that solutions to a fractional stochastic heat equation driven by colored Gaussian noise stay close to zero, providing estimates for small ball probabilities under specific conditions.

Contribution

It offers new estimates for small ball probabilities of solutions to a fractional stochastic heat equation with colored noise, extending previous results to this setting.

Findings

Derived bounds for small ball probabilities of the solution

Extended analysis to fractional Laplacian with  in (1,2]

Handled colored spatial noise with Gaussian properties

Abstract

We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})= -(-\Delta)^{\alpha/2}u(t,\textbf{x})+\sigma(t,\textbf{x},u)\dot{F}(t,\textbf{x})\quad \textbf{x}\in \mathbb{T}^d,t\in\mathbb{R}_+ ,\] where $\alpha\in(1,2]$ and $\dot{F}(t,\textbf{x})$ 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 $u(0,\textbf{x})\equiv 0$.