On the Besov Regularity of Periodic L\'evy Noises

TL;DR

This paper investigates the Besov regularity of Le9vy white noises on the torus, providing foundational results for understanding their rough sample paths and wavelet approximation properties.

Contribution

It establishes the well-posedness of Besov regularity measurement for Le9vy noises and offers detailed regularity results for specific subclasses like compound Poisson and symmetric-b1b6-stable noises.

Findings

Proves Besov spaces are in the cylindrical b5-field of generalized functions.

Provides regularity results for general Le9vy white noises.

Characterizes wavelet approximation properties of these stochastic processes.

Abstract

In this paper, we study the Besov regularity of L\'evy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general L\'evy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$\alpha$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $\sigma$-field of the space of generalized functions. These results pave the way to the characterization of the $n$-term wavelet approximation properties of stochastic processes.