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

Abstract

We consider the solution of ${\partial }_{t}u={\partial }_x^{2}u+{\partial }_x{\partial }_{t}B,(x,t)\in R\times (0,\infty )$, subject to $u(x,0)=0,x\in R$, 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 $R\times (0,\infty )$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\in R^2$ 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.

arXiv comment36 pages