On exact Hausdorff measure functions of operator semistable L\'evy processes

TL;DR

This paper determines the exact Hausdorff measure functions for the range of operator semistable Lévy processes, providing precise geometric measure results and asymptotic bounds under spectral assumptions.

Contribution

It introduces exact Hausdorff measure functions for the range of operator semistable Lévy processes and derives related Tauberian and asymptotic results.

Findings

Exact Hausdorff measure functions for the process range

Tauberian results for semistable subordinators

Sharp bounds for expected sojourn times

Abstract

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process on $\mathbb{R}^d$ with exponent $E$, where $E$ is an invertible linear operator on $\mathbb{R}^d$. In this paper we determine exact Hausdorff measure functions for the range of $X$ over the time interval $[0,1]$ under certain assumptions on the principal spectral component of $E$. As a byproduct we also present Tauberian results for semistable subordinators and sharp bounds for the asymptotic behavior of the expected sojourn times of $X$.