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

Abstract

We define various higher-order Markov properties for stochastic processes $(X(t))_{t\in \mathbb{T}}$, indexed by an interval $\mathbb{T} \subseteq \mathbb{R}$ 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 $\mathcal{L} X = \dot W^Q\!$, where $\mathcal{L}$ is a linear operator acting on functions mapping from $\mathbb{T}$ to $U$ and $(\dot W^Q(t))_{t\in\mathbb{T}}$ is the formal derivative of a $U$-valued (cylindrical) $Q$-Wiener process, we prove necessary and sufficient conditions for the weakest Markov property via locality of the precision operator $\mathcal{L}^*\! \mathcal{L}$. As an application, we consider the space-time fractional parabolic operator $\mathcal{L} = (\partial_t + A)^\gamma$ 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 $\gamma = N \in \mathbb{N}$ and show that a necessary condition for the weakest Markov property is generally not satisfied if $\gamma \notin \mathbb{N}$. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle-Mat\'ern Gaussian random fields if $U = L^2(\mathcal{D})$ for a spatial domain $\mathcal{D}\subseteq\mathbb{R}^d\!$. 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 $\mathcal{L} = (\partial_t + \varepsilon \, \mathrm{Id}_U)^{H + \frac{1}{2}}$ for $\varepsilon \downarrow 0$.