Hilbert--Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations

TL;DR

This paper establishes Hilbert--Schmidt regularity estimates for symmetric integral operators on bounded domains, linking noise regularity with differential operator properties, with applications to SPDEs and their numerical approximations.

Contribution

It introduces new Hilbert--Schmidt norm estimates for integral operators with symmetric kernels, applicable to SPDEs and including general Schatten class results for homogeneous kernels.

Findings

Derived Hilbert--Schmidt estimates for integral operators on bounded domains.

Connected noise regularity with differential operator properties in SPDEs.

Provided Schatten class norm results for homogeneous kernels like Matérn kernels.

Abstract

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert--Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert--Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Mat\'ern kernels.