On the spatial dynamics of the solution to the stochastic heat equation

Abstract

We consider the solution of $\partial_t u=\partial_x^2u+\partial_x\partial_t B,\,(x,t)\in\mathbb{R}\times(0,\infty)$, subject to $u(x,0)=0,\,x\in\mathbb{R}$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\partial_x^2 u +[\,( \partial_t^2)^{1/2}+\sqrt{2}\partial_x( \partial_t^2)^{1/4}\,]\,u^a=\partial_x\partial_t{\tilde B}$ in $\mathbb{R}\times(0,\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\in\mathbb{R}^2$ which is antisymmetric in $t$ and $\tilde{B}$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,\partial_x u)$ when seen as a process indexed by $x\ge x_0$, $x_0$ fixed, taking values in a state space of functions in $t$. The method of proof is based on enlargement of filtration and we discuss how our method could be applied to other quasi-linear SPDEs.