Small ball probabilities and a support theorem for the stochastic heat equation

Abstract

We consider the following stochastic partial differential equation on $\mathit{t}\ge 0,\mathit{x}\in [0,\mathit{J}],\mathit{J}\ge 1$, where we consider $[0,\mathit{J}]$ to be the circle with end points identified,

$${\partial }_{\mathit{t}}\mathbf{u}(\mathit{t},\mathit{x})=\frac{1}{2}{\partial }_{\mathit{x}}^2\mathbf{u}(\mathit{t},\mathit{x})+\mathbf{g}(\mathit{t},\mathit{x},\mathbf{u})+\mathit{\sigma }(\mathit{t},\mathit{x},\mathbf{u})\dot{\mathbf{W}}(\mathit{t},\mathit{x}),$$

$\dot{\mathbf{W}}(\mathit{t},\mathit{x})$ is 2-parameter d-dimensional vector valued white noise and σ is function from ${\mathbb{R}}_{+}\times \mathbb{R}\times {\mathbb{R}}^{\mathit{d}}$ to space of symmetric $\mathit{d}\times \mathit{d}$ matrices which is Lipschitz in u. We assume that σ is uniformly elliptic and that g is uniformly bounded. Assuming that $\mathbf{u}(0,\mathit{x})\equiv \mathbf{0}$, we prove small ball probabilities for the solution u. We also prove a support theorem for solutions, when $\mathbf{u}(0,\mathit{x})$ is not necessarily zero.