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 ${\mathcal{L}}^{\ast }\mathcal{L}$. 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-Matérn 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 $\mathcal{L}=({\partial }_t+\epsilon {\mathrm{Id}}_U{)}^{H+\frac{1}{2}}$ for $\epsilon \downarrow 0$.

arXiv comment39 pages, 1 figure

References in corpus (7)

• Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
• The rational SPDE approach for Gaussian random fields with general smoothness
• Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations
• Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
• Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle-Matérn fields
• Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
• Regularity theory for a new class of fractional parabolic stochastic evolution equations

Cited by in corpus (1)

• Spectral approximation of a new class of stochastic fractional evolution equations