Regularisation of cylindrical L\'evy processes in Besov spaces

TL;DR

This paper investigates the regularity of cylindrical Lévy processes in Besov spaces by characterizing Lévy measures and identifying spaces where these processes can be regularized as classical stochastic processes.

Contribution

It introduces a method to quantify the irregularity of cylindrical Lévy processes in Besov spaces and characterizes spaces allowing regularized versions of these processes.

Findings

Identifies Besov spaces where cylindrical Lévy processes can be regularized.

Characterizes Lévy measures on Besov spaces.

Determines conditions for Radonifying embeddings of $L^2(\mathbb{R}^d)$ into Besov spaces.

Abstract

In this work, we quantify the irregularity of a given cylindrical L\'evy process $L$ in $L^2({\mathbb R}^d)$ by determining the range of weighted Besov spaces $B$ in which $L$ has a regularised version $Y$, that is a stochastic process $Y$ in the classical sense with values in $B$. Our approach is based on characterising L\'evy measures on Besov spaces. As a by-product, we determine those Besov spaces $B$ for which the embedding of $L^2({\mathbb R}^d)$ into $B$ is $0$-Radonifying and $p$-Radonifying for $p>1$.