A support theorem for parabolic stochastic PDEs with nondegenerate H\"older diffusion coefficients

TL;DR

This paper establishes a support theorem for a class of parabolic stochastic PDEs with non-Lipschitz, H"older continuous diffusion coefficients, extending previous results that required Lipschitz continuity.

Contribution

It introduces new support theorems and small ball probability estimates for SPDEs with H"older continuous diffusion coefficients, broadening the scope of existing support theorems.

Findings

Support theorems for SPDEs with H"older continuous diffusion coefficients

Sharp two-sided estimates of stochastic integrals used

Extension of support results beyond Lipschitz conditions

Abstract

In this paper we work with parabolic SPDEs of the form $$ \partial_t u(t,x)=\partial_x^2 u(t,x)+g(t,x,u)+\sigma(t,x,u)\dot{W}(t,x) $$ with Neumann boundary conditions, where $x\in[0,1]$, $\dot{W}(t,x)$ is the space-time white noise on $(t,x)\in[0,\infty)\times [0,1]$, $g$ is uniformly bounded, and the solution $u\in\mathbb{R}$ is real valued. The diffusion coefficient $\sigma$ is assumed to be uniformly elliptic but only H\"older continuous in $u$. Previously, support theorems for SPDEs have only been established assuming that $\sigma$ is Lipschitz continuous in $u$. We obtain new support theorems and small ball probabilities in this $\sigma$ H\"older continuous case via the recently established sharp two sided estimates of stochastic integrals.