Uniform Hausdorff dimension result for the inverse images of stable L\'evy processes

TL;DR

This paper proves a uniform Hausdorff dimension result for inverse images of certain stable Lévy processes, extending Kaufman's theorem for Brownian motion using new covering principles for Markov processes.

Contribution

It introduces a novel method based on covering principles to establish Hausdorff dimension results for inverse images of stable Lévy processes, extending previous work on Brownian motion.

Findings

Establishes a uniform Hausdorff dimension result for inverse images of stable Lévy processes.

Extends Kaufman's theorem from Brownian motion to a broader class of Lévy processes.

Develops new covering principles for Markov processes to achieve these results.

Abstract

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha$-stable L\'evy processes with $1< \alpha\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from that of Kaufman and depends on covering principles for Markov processes.