Small ball probability estimates for the Hölder semi-norm of the stochastic heat 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 Hölder semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these Hölder semi-norms.