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 (X(t))_t∈T, indexed by an interval T⊆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=Ẇ, where L is a linear operator acting on functions mapping from T to U and (Ẇ(t))_t∈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^∗L. As an application, we consider the space–time fractional parabolic operator L=(∂_t+A)^γ of order γ∈(1/2,∞), where −A is a linear operator generating a C₀-semigroup on U. We prove that the resulting solution process satisfies an Nth order Markov property if γ=N∈N and show that a necessary condition for the weakest Markov property is generally not satisfied if γ∉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=L²(D) for a spatial domain D⊆R^d. Secondly, we show that a U-valued analog to the fractional Brownian motion with Hurst parameter H∈(0,1) can be obtained as the limiting case of [Formula presented] for ɛ↓0.