A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients

Abstract

In this paper we work with parabolic SPDEs of the form $${\partial }_{t}u(t,x)={\partial }_x^{2}u(t,x)+g(t,x,u)+\ddot{I}ƒ(t,x,u)\dot{W}(t,x)$$ 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 $\ddot{I}ƒ$ is assumed to be uniformly elliptic but only Hölder continuous in $u$. Previously, support theorems for SPDEs have only been established assuming that $\ddot{I}ƒ$ is Lipschitz continuous in $u$. We obtain new support theorems and small ball probabilities in this $\ddot{I}ƒ$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals.

12 pages. To appear in Stochastics and Partial Differential Equations: Analysis and Computations