Multiple points of operator semistable L\'evy processes

TL;DR

This paper determines the Hausdorff dimension and existence conditions of k-multiple points for symmetric operator semistable Lévy processes, extending previous results from the stable case to the semistable case for all k ≥ 2.

Contribution

It provides a formula for the Hausdorff dimension of k-multiple points based on eigenvalues of the stability exponent and establishes necessary and sufficient conditions for their existence.

Findings

Hausdorff dimension expressed via eigenvalues of the stability exponent

Necessary and sufficient conditions for the existence of k-multiple points

Extension of previous double point results to all k ≥ 2

Abstract

We determine the Hausdorff dimension of $k$-multiple points for a symmetric operator semistable L\'evy process $X=\{X(t), t\in\mathbb{R}_+\}$ in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of $k$-multiple points. Our results extend to all $k\geq2$ the recent work [23], where the set of double points $(k = 2)$ was studied in the symmetric operator stable case.