When does a Gaussian process have its paths in a reproducing kernel Hilbert space?

TL;DR

This paper characterizes when Gaussian processes have paths contained in reproducing kernel Hilbert spaces, providing both positive and negative results for various classical and Sobolev space-related Gaussian processes.

Contribution

It introduces a new criterion to exclude the existence of RKHSs containing Gaussian process paths and fully characterizes such existence for several classical families.

Findings

Negative results exclude RKHSs for many Gaussian processes.

Positive results identify conditions under which RKHSs contain process paths.

Complete characterizations for classical Gaussian families and Sobolev space-related processes.

Abstract

We investigate for which Gaussian processes there do or do not exist reproducing kernel Hilbert spaces (RKHSs) that contain almost all of their paths. In particular, we establish a new result that makes it possible to exclude the existence of such RKHSs in many cases. Moreover, we combine this negative result with some known techniques to establish positive results. Here it turns out that for many classical families of Gaussian processes we can fully characterize for which members of these families there exist RKHSs containing the paths. Similar characterizations are obtained for Gaussian processes, for which the RKHSs of their covariance functions are Sobolev spaces or Sobolev spaces of mixed smoothness.