Multifractal analysis of L\'evy fields

TL;DR

This paper investigates the local regularity of Levy fields, generalizing Levy processes to multiple dimensions, by analyzing their singularity spectrum and intersection properties of their Hölder singularity sets.

Contribution

It determines the spectrum of singularities for Levy fields and proves their Hölder singularity sets have a large intersection property, extending previous one-dimensional results.

Findings

Determined the spectrum of singularities for Levy fields.

Proved the large intersection property of Hölder singularity sets.

Extended multifractal analysis to multivariate Levy fields.

Abstract

We study the pointwise regularity properties of the L\'evy fields introduced by T. Mori; these fields are the most natural generalization of L\'evy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their H\"older singularity sets satisfy a large intersection property in the sense of K. Falconer.