Regularity of Gaussian Processes on Dirichlet spaces

TL;DR

This paper investigates the regularity of Gaussian processes on compact metric spaces with a Dirichlet structure, linking the process regularity to the covariance's Besov regularity, especially on homogeneous spaces like spheres.

Contribution

It establishes a novel equivalence between Gaussian process regularity and covariance regularity under Dirichlet structures with commuting operators.

Findings

Regularity of Gaussian processes is characterized by covariance Besov regularity.

Results apply to processes on homogeneous spaces and spheres.

Provides a framework for understanding process regularity via Dirichlet structures.

Abstract

We are interested in the regularity of centered Gaussian processes (Z_x), x in M indexed by compact metric spaces M. It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x,y) = E(Z_xZ_y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x,y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.