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 $u_{t}=\Delta u+ \dot F, u(0,\cdot)=0$ driven by a 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 $\alpha=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $\alpha=4k,k \in \bN_{+}$.