  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.

Probability