Multifractal analysis for the occupation measure of stable-like processes

TL;DR

This paper studies the multifractal spectrum of occupation measures of stable-like processes, revealing a random spectrum that varies with trajectories, contrasting with deterministic spectra in other processes.

Contribution

It introduces new methods to analyze the multifractal spectrum of occupation measures for stable-like processes, showing its randomness and unique shape.

Findings

Multifractal spectrum is random and trajectory-dependent.

The spectrum's shape is highly original and reflects diverse local behaviors.

New techniques provide fine estimates on Hausdorff dimensions of jump configurations.

Abstract

In this article, we investigate the local behaviors of the occupation measure $\mu$ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A $\subset$ R by the sample paths of M. We compute the multifractal spectrum of $\mu$, which turns out to be random, depending on the trajec-tory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as L{\'e}vy processes), since the multifractal spectrum is usually determinis-tic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behaviors. 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.