Small ball probability estimates for the H\"older semi-norm of the stochastic heat equation

TL;DR

This paper derives near optimal bounds for small ball probabilities of the H"older semi-norms of solutions to the stochastic heat equation with periodic boundary conditions driven by space-time white noise, and proves a support theorem in these norms.

Contribution

It provides new estimates on small ball probabilities for the H"older semi-norms of the stochastic heat equation solutions and establishes a support theorem in these norms.

Findings

Near optimal bounds on small ball probabilities for H"older semi-norms

Support theorem in H"older semi-norms for the stochastic heat equation

Analysis under natural conditions for the equation

Abstract

We consider the stochastic heat equation on $[0,\,1]$ with periodic boundary conditions and driven by space-time white noise. Under various natural conditions, we study small ball probabilities for the H\"older semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these H\"older semi-norms.