Path regularity of Brownian motion and Brownian sheet

TL;DR

This paper reviews classical and recent results on the regularity of Brownian motion and Brownian sheet paths, introducing new Besov-type spaces where these paths almost surely reside.

Contribution

It proposes new smaller Besov-type spaces for Brownian paths and extends regularity results to the multivariate Brownian sheet with dominating mixed smoothness.

Findings

Brownian motion paths satisfy certain Besov regularity almost surely.

New Besov-type spaces are identified where paths lie almost surely.

Extensions to the Brownian sheet and mixed smoothness spaces are provided.

Abstract

By the work of P. L\'evy, the sample paths of the Brownian motion are known to satisfy a certain H\"older regularity condition almost surely. This was later improved by Ciesielski, who studied the regularity of these paths in Besov and Besov-Orlicz spaces. We review these results and propose new function spaces of Besov type, strictly smaller than those of Ciesielski and L\'evy, where the sample paths of the Brownian motion lie in almost surely. In the same spirit, we review and extend the work of Kamont, who investigated the same question for the multivariate Brownian sheet and function spaces of dominating mixed smoothness.