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

Abstract

In this paper we work with parabolic SPDEs of the form ∂tu(t,x)=∂x2u(t,x)+g(t,x,u)+σ(t,x,u)W˙(t,x)$$\begin{aligned} \partial _t u(t,x)=\partial _x^2 u(t,x)+g(t,x,u)+\sigma (t,x,u)\dot{W}(t,x) \end{aligned}$$with Neumann boundary conditions, where x∈[0,1]$$x\in [0,1]$$, W˙(t,x)$$\dot{W}(t,x)$$ is the space-time white noise on (t,x)∈[0,∞)×[0,1]$$(t,x)\in [0,\infty )\times [0,1]$$, g is uniformly bounded, and the solution u∈R$$u\in \mathbb {R}$$ is real valued. The diffusion coefficient σ$$\sigma $$ is assumed to be uniformly elliptic but only Hölder continuous in u. Previously, support theorems for SPDEs have only been established assuming that σ$$\sigma $$ is Lipschitz continuous in u. We obtain new support theorems and small ball probabilities in this σ$$\sigma $$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals.