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 Levy 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.
Highlights
The occupation measure of a Rd-valued stochastic process (Xt)t≥0 describes the time spent by X in any borelian set A ⊂ Rd
We describe the local behavior of this occupation measure via its multifractal analysis
Multifractal analysis is identified as a fruitful approach to provide organized information on the fluctuation of the local regularity of functions and measures, see for instance [18, 13]
Summary
The occupation measure of a Rd-valued stochastic process (Xt)t≥0 describes the time spent by X in any borelian set A ⊂ Rd. We obtain the almost-sure multifractal spectrum of the occupation measure of stable-like jump diffusions, 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. 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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