Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

We consider Gaussian measures $mu, tilde{mu}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2beta}$ resp. $tilde{A}^{-2tilde{beta}}$, and derive necessary and sufficient conditions on $A, tilde{A}$ and $beta, tilde{beta} > 0$ for I. equivalence of the measures $mu$ and $tilde{mu}$, and II. uniform asymptotic optimality of linear predictions for $mu$ based on the misspecified measure $tilde{mu}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Matern Gaussian random fields, where $A$ and $tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $mathcal{D}subsetmathbb{R}^d$ 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-Matern fields.