Uniform dimension results for the inverse images of symmetric L\'evy processes

TL;DR

This paper establishes uniform Hausdorff and packing dimension results for inverse images of symmetric Lévy processes, extending classical results for Brownian motion and stable processes, and also provides bounds for local times' modulus of continuity.

Contribution

It extends dimension results to a broad class of symmetric Lévy processes and introduces bounds for local times' uniform modulus of continuity.

Findings

Hausdorff and packing dimension results for inverse images

Extension of Kaufman's and Song et al.'s results to broader Lévy processes

Upper bounds for local times' uniform modulus of continuity

Abstract

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric L\'evy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for $\alpha$-stable L\'evy processes with $1<\alpha<2$. Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.