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.

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

We formulate standard and multilevel Monte Carlo methods for the kth moment M_ε^k[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗_ε^kE. For the standard Monte Carlo estimator of M_ε^k[ξ], we prove the k-independent convergence rate [Formula presented] in the L_q(Ω;⊗_ε^kE)-norm, provided that (i) ξ∈L_kq(Ω;E) and (ii) q∈[p,∞), where p∈[1,2] is the Rademacher type of E. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the L_q(Ω;⊗_ε^kE)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space E is p=2, our findings coincide with known results for Hilbert space valued random variables. We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type p<2, are indicated.

We consider two Gaussian measures μ, ˜μ on a separable Hilbert space, with fractional-order covariance operators A^−2β and Ã^−2˜β, respectively, and derive necessary and sufficient conditions on A, à and β, ˜β > 0 for I. equivalence of the measures μ and ˜μ, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure ˜μ. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and à are elliptic second-order differential operators, formulated on a bounded Euclidean domain D ⊂ R^d and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

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.

Optimal linear prediction (aka. kriging) of a random field {Z(x)} x∈X indexed by a compact metric space (X, dX ) can be obtained if the mean value function m: X →R and the covariance function ∂: X × X →R of Z are known. We consider the problem of predicting the value of Z(x*) at some location x*∈ X based on observations at locations {xj }nj =1, which accumulate at x*as n→∞(or, more generally, predicting φ(Z) based on {φj (Z)}nj =1 for linear functionals φ,φ1, . . . , φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m, ∂), without any restrictive assumptions on ,∂ ∂ such as stationarity.We, for the first time, provide necessary and sufficient conditions on (m,∂) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X ⊂ Rd with Matérn or periodic covariance functions, and on the sphere X = S2 for the case of two isotropic covariance functions.

We analyze several types of Galerkin approximations of a Gaussian random field Z: D× Ω→ R indexed by a Euclidean domain D⊂ R^d whose covariance structure is determined by a negative fractional power L^-2β of a second-order elliptic differential operator L: = - ∇ · (A∇) + κ². Under minimal assumptions on the domain D, the coefficients A: D→ R^d×d, κ: D→ R, and the fractional exponent β> 0 , we prove convergence in L_q(Ω; H^σ(D)) and in L_q(Ω; C^δ(D¯)) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H^1+α(D) -regularity of the differential operator L, where 0 < α≤ 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L_∞(D× D) and in the mixed Sobolev space H^σ,σ(D× D) , showing convergence which is more than twice as fast compared to the corresponding L_q(Ω; H^σ(D)) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where A≡IdRd and κ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in d= 2 dimensions, where A: D→ R^{2 × 2} and κ: D→ R are spatially varying.