Open Access
2017 Multifractal analysis for the occupation measure of stable-like processes
Stéphane Seuret, Xiaochuan Yang
Electron. J. Probab. 22: 1-36 (2017). DOI: 10.1214/17-EJP48


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.


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Stéphane Seuret. Xiaochuan Yang. "Multifractal analysis for the occupation measure of stable-like processes." Electron. J. Probab. 22 1 - 36, 2017.


Received: 23 June 2016; Accepted: 9 March 2017; Published: 2017
First available in Project Euclid: 30 May 2017

zbMATH: 1364.60075
MathSciNet: MR3661661
Digital Object Identifier: 10.1214/17-EJP48

Primary: 28A78 , 28A80 , 60H10 , 60J25 , 60J75

Keywords: Hausdorff measure and dimension , Markov and Lévy processes , occupation measure

Vol.22 • 2017
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