Small ball probabilities and a support theorem for the stochastic heat equation

TL;DR

This paper establishes small ball probability estimates and a support theorem for solutions to a stochastic heat equation with multiplicative noise on a circular domain, under certain regularity and ellipticity conditions.

Contribution

It provides the first known small-ball probability bounds and a support theorem for the stochastic heat equation with multiplicative noise on a circle.

Findings

Proved small-ball probability estimates for the solution.

Established a support theorem for the stochastic heat equation.

Demonstrated results under uniform ellipticity and boundedness conditions.

Abstract

We consider the following stochastic partial differential equation on $t \geq 0, x\in[0,J], J \geq 1$ where we consider $[0,J]$ to be the circle with end points identified: \begin{equation*} \partial_t{\mathbf u}(t,x) =\frac{1}{2}\,\partial_x^2 {\mathbf u}(t,x) + {\mathbf g}(t,x,\mathbf u) + {\mathbf \sigma}(t,x, {\mathbf u})\dot {\mathbf W}(t,x) , \end{equation*} and $\dot {\mathbf W }(t,x)$ is 2-parameter $d$-dimensional vector valued white noise and ${\mathbf \sigma}$ is function from ${\mathbb R}_+\times {\mathbb R} \times {\mathbb R}^d \rightarrow {\mathbb R}^d$ to space of symmetric $d\times d$ matrices which is Lipschitz in $\mathbf u$. We assume that $\sigma$ is uniformly elliptic and that $\mathbf g$ is uniformly bounded. Assuming that ${\mathbf u}(0,x) \equiv \mathbf 0$, we prove small-ball probabilities for the solution $\mathbf u$. We also prove a support theorem for solutions, when ${\mathbf u}(0,x)$ is not necessarily zero.