Sample Paths of White Noise in Spaces with Dominating Mixed Smoothness

TL;DR

This paper demonstrates that white noise sample paths belong to Besov spaces with dominating mixed smoothness, showing they are smoother than previously thought, and applies these results to boundary noise in PDEs.

Contribution

It introduces a novel analysis of white noise regularity in mixed smoothness Besov spaces, contrasting with isotropic space results, and extends to PDE boundary noise regularity.

Findings

White noise sample paths are in Besov spaces with dominating mixed smoothness.

White noise is smoother than in isotropic spaces.

New regularity results for PDE solutions with boundary noise.

Abstract

The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.