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

In this paper we work with parabolic SPDEs of the form $${\mathrm{\partial }}_{t}u(t,x)={\mathrm{\partial }}_x^{2}u(t,x)+g(t,x,u)+\sigma (t,x,u)\dot{W}(t,x)$$ with Neumann boundary conditions, where $x\in [0,1]$, $\dot{W}(t,x)$ is the space-time white noise on $(t,x)\in [0,\mathrm{\infty })\times [0,1]$, $g$ 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 Hölder 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 $\sigma$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals.