Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations

Abstract

We define a number of higher-order Markov properties for stochastic processes ${\left(X\left(t\right)\right)}_{t\in T}$, indexed by an interval $T\subseteq R$ and taking values in a real and separable Hilbert space $U$. We furthermore investigate the relations between them. In particular, for solutions to the stochastic evolution equation $LX=\dot{W}$, where $L$ is a linear operator acting on functions mapping from $T$ to $U$ and ${\left(\dot{W}\left(t\right)\right)}_{t\in T}$ is the formal derivative of a $U$-valued cylindrical Wiener process, we prove necessary and sufficient conditions for the weakest Markov property via locality of the precision operator $L^{\ast }L$.

As an application, we consider the space–time fractional parabolic operator $L={({\partial }_t+A)}^{\gamma }$ of order $\gamma \in (1/2,\infty )$, where $-A$ is a linear operator generating a $C_0$-semigroup on $U$. We prove that the resulting solution process satisfies an $N$th order Markov property if $\gamma =N\in N$ and show that a necessary condition for the weakest Markov property is generally not satisfied if $\gamma \notin N$. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle–Matérn Gaussian random fields if $U=L^2\left(D\right)$ for a spatial domain $D\subseteq R^d$. Secondly, we show that a $U$-valued analog to the fractional Brownian motion with Hurst parameter $H\in (0,1)$ can be obtained as the limiting case of $L={({\partial }_t+\varepsilon {\mathrm{Id}}_U)}^{H+\frac{1}{2}}$ for $\varepsilon \downarrow 0$.