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

Abstract

We consider the following stochastic partial differential equation on $t\ge 0,x\in [0,J],J\ge 1$ where we consider $[0,J]$ to be the circle with end points identified: \begin{equation*} \partial_t{\mathbf u}(t,x) =\frac{1}{2}\,\partial_x^2 {\mathbf u}(t,x) + {\mathbf g}(t,x,\mathbf u) + {\mathbf σ}(t,x, {\mathbf u})\dot {\mathbf W}(t,x) , \end{equation*} and $\dot{\mathbf{W}}(t,x)$ is 2-parameter $d$-dimensional vector valued white noise and $\ddot{\mathbf{I}}ƒ$ is function from ${\mathbb{R}}_{+}\times \mathbb{R}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ to space of symmetric $d\times d$ matrices which is Lipschitz in $\mathbf{u}$. We assume that $\ddot{I}ƒ$ is uniformly elliptic and that $\mathbf{g}$ is uniformly bounded. Assuming that $\mathbf{u}(0,x)\equiv 0$, we prove small-ball probabilities for the solution $\mathbf{u}$. We also prove a support theorem for solutions, when $\mathbf{u}(0,x)$ is not necessarily zero.