The stochastic heat equation driven by a Gaussian noise: germ Markov property

Abstract

Let $u = {u(t,x); t \in [0,T],x \in\mathbb{R}^d}$ be the process solution of the stochastic heat equation $u_t = \Delta u +\dot{F},\, u(o,\cdot)=0$ driven by Gaussian noise $\dot{f}$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel Kernel of order $a=2k,\, k\in\mathbb{N}_+$, or $f$ is the Riesz kernel of order $\alpha =4k,\, k\in\mathbb{N}_+$.