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.

Let the abstract fractional space–time operator (∂t+A)s be given, where s∈(0,∞) and -A:D(A)⊆X→X is a linear operator generating a uniformly bounded strongly measurable semigroup (S(t))t≥0 on a complex Banach space X. We consider the corresponding Dirichlet problem of finding u:R→X such that (Formula presented.) for given t0∈R and g:(-∞,t0]→X. We define the concept of Lp-solutions, to which we associate a mild solution formula which expresses u in terms of g and (S(t))t≥0 and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem u′+Au=0 on (t0,∞) with u(t0)=x∈D(A)¯. Moreover, we include a comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems.

A new class of fractional-order parabolic stochastic evolution equations of the form (∂t+A) ^γX(t)=W˙ ^Q(t), t∈[0,T], γ∈(0,∞), is introduced, where -A generates a C ₀-semigroup on a separable Hilbert space H and the spatiotemporal driving noise W˙ ^Q is the formal time derivative of an H-valued cylindrical Q-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when A:=L ^β and Q:=L~ ^-α are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle–)Matérn fields to space–time.