Sobolev regularity of Gaussian random fields

TL;DR

This paper characterizes Gaussian processes with sample paths in Sobolev spaces by analyzing the Sobolev regularity of their covariance functions and related integral operators, linking these properties to Mercer decompositions.

Contribution

It provides a complete characterization of Gaussian processes with Sobolev regularity of sample paths based on covariance function properties and integral operator decompositions.

Findings

Characterization of Gaussian processes with Sobolev sample path regularity.

Connection between covariance function regularity and Mercer decompositions.

Extension of existing results on Sobolev regularity of Gaussian processes.

Abstract

In this article, we fully characterize the measurable Gaussian processes $(U(x))_{x\in\mathcal{D}}$ whose sample paths lie in the Sobolev space of integer order $W^{m,p}(\mathcal{D}),\ m\in\mathbb{N}_0,\ 1 <p<+\infty$, where $\mathcal{D}$ is an arbitrary open set. The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case $p=2$, additional links are made w.r.t. the Mercer decompositions of the said integral operators, their trace and the imbedding of the RKHS in $W^{m,2}(\mathcal{D})$. We provide simple examples and partially recover recent results pertaining to the Sobolev regularity of Gaussian processes.