Large deviations for a class of stochastic partial differential equations

Abstract

We consider the random fields $X^{\varepsilon}(t, q), \ t\geq 0, \ q\in {\mathcal O},$ goverened by stochastic partial differential equations driven by a Gaussian white noise in space-time, where $\mathcal O$ is a bounded domain in ${\mathbb R}^d$ with regular boundary. To study the continuity of the random fields $X^\varepsilon$ in space and time variables, we prove an analogue of Garsia's theorem. We then derive the large deviation results based on the methods used by the second author in another paper. This article provides an alternative proof of Sower's result for the case of d = 1.