Regularity of Gaussian white noise on the d-dimensional torus

TL;DR

This paper establishes the optimal regularity properties of Gaussian white noise on the d-dimensional torus within Besov and Fourier-Besov spaces, demonstrating precise path regularity results.

Contribution

It proves the optimal regularity of Gaussian white noise paths in Besov and Fourier-Besov spaces on the torus, extending understanding of noise regularity in these function spaces.

Findings

Gaussian white noise paths are in $B^{-d/2}_{p, olinebreak \\infty}$ spaces, for $p \\in [1, \\infty)$

Paths are also in Fourier-Besov spaces $\\hat{b}^{-d/p}_{p, olinebreak \\infty}$

Results are shown to be optimal in multiple ways

Abstract

In this paper we prove that a Gaussian white noise on the $d$-dimensional torus has paths in the Besov spaces $B^{-d/2}_{p,\infty}(\T^d)$ with $p\in [1, \infty)$. This result is shown to be optimal in several ways. We also show that Gaussian white noise on the $d$-dimensional torus has paths in a the Fourier-Besov space $\hat{b}^{-d/p}_{p,\infty}(\T^d)$. This is shown to be optimal as well.