Stochastic heat equation with Ornstein–Uhlenbeck operator

Abstract

We study the stochastic partial differential equation ∂u∂t=Lu+uẆ in Skorohod sense, where L is the generator of a one-dimensional Ornstein–Uhlenbeck (OU) process X and Ẇ is a (1+1)-dimensional space–time white noise. We follow the method established in Hu and Nualart (2009) to prove a Feynman–Kac type representation of the moments of solution. Through this representation, we use the method established in Chen (2015) and a large deviation principle of OU process to obtain an intermittency result.