Multifractal analysis for the occupation measure of stable-like processes

Abstract

In this article, we investigate the local behavior of the occupation measure $\mu $ of a class of real-valued Markov processes $\mathcal{M} $, defined via a SDE. This (random) measure describes the time spent in each set $A\subset \mathbb{R} $ by the sample paths of $\mathcal{M} $. We compute the multifractal spectrum of $\mu $, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behavior. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.