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.

In this thesis, we study deterministic and stochastic abstract evolution equations which are of fractional order, either spatially or spatiotemporally.

The preliminary concepts necessary to interpret such equations are collected in the first chapter.

The second chapter is devoted to analyzing a class of equations which generalizes the spatial Whittle–Matérn stochastic partial differential equations (SPDEs) to space–time. We define solution concepts, establish their well-posedness and equivalence, and study their spatiotemporal regularity and covariance structure. The abstract spatial operator in these equations is assumed to be the negative generator of a strongly continuous semigroup on a separable Hilbert space; for certain spatial regularity results, it is moreover assumed to be analytic.

In the third chapter, we define various higher-order Markov properties for Hilbert-space-valued stochastic processes and investigate the relations between them. For solutions to linear abstract SPDEs, we identify two sets of additional conditions under which locality of the precision operator is either necessary or sufficient to satisfy the weakest Markov property. We show that the mild solutions from Chapter 2 satisfy a higher-order Markov property if the orders of the equations are non-fractional (i.e., integer) and that, conversely, a necessary condition for the weakest Markov property is, in general, not satisfied if this parameter is fractional. We moreover establish that an infinite-dimensional analog of the fractional Brownian motion can be obtained as a limiting case of this class of equations.

The fourth chapter concerns the deterministic natural Dirichlet problem for nonlocal abstract space–time operators posed on an arbitrary Banach space. We impose that the whole "past" of the solution equals a given function, i.e., its values are prescribed at all times preceding some (non-trivial) initial time. If the semigroup associated to the spatial operator is exponentially stable, we show that the problem is well-posed in an L^p-sense with p in [1, infinity]. Whenever such solutions are continuous, they satisfy a mild solution formula which expresses them in terms of the initial data and the semigroup, thus generalizing the well-known variation of constants formula for the first-order abstract Cauchy problem. A comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems is included.

Finally, in the fifth chapter, we study the convergence of a sequence of semilinear parabolic stochastic evolution equations posed on a sequence of Banach spaces approximating a limiting space. The abstract "discrete-to-continuum approximation" setting is encoded using projection and lifting operators. These allow us to define and compare the discretized equations, and to formulate conditions under which their solutions are well posed and convergent when lifted to a common state space. Our framework is applied to the case where the limiting problem is an SPDE whose linear part is a generalized Whittle–Matérn operator on a manifold, discretized by a sequence of geometric graphs constructed from a (random) point cloud.

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.