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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 thespace-time white noise on $(t,x)\in[0,\infty)\times [0,1]$, $g$ is uniformlybounded, and the solution $u\in\mathbb{R}$ is real valued. The diffusioncoefficient $\sigma$ is assumed to be uniformly elliptic but only H\"oldercontinuous in $u$. Previously, support theorems for SPDEs have only beenestablished assuming that $\sigma$ is Lipschitz continuous in $u$. We obtainnew support theorems and small ball probabilities in this $\sigma$ H\"oldercontinuous case via the recently established sharp two sided estimates ofstochastic integrals.

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