Hausdorff dimension of operator semistable L\'evy processes

TL;DR

This paper determines the Hausdorff dimension of the image of Borel sets under operator semistable Lévy processes, extending previous results from the selfsimilar case by analyzing eigenvalues of the exponent operator.

Contribution

It generalizes the Hausdorff dimension results from operator stable to operator semistable Lévy processes using refined analytical techniques.

Findings

Explicit formula for Hausdorff dimension of X(B) in terms of eigenvalues of E.

Extension of previous selfsimilar results to semistable processes.

Provides a method to compute dimensions for arbitrary Borel sets.

Abstract

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process in $\rd$ with exponent $E$, where $E$ is an invertible linear operator on $\rd$ and $X$ is semi-selfsimilar with respect to $E$. By refining arguments given in Meerschaert and Xiao \cite{MX} for the special case of an operator stable (selfsimilar) L\'evy process, for an arbitrary Borel set $B\subseteq\rr_+$ we determine the Hausdorff dimension of the partial range $X(B)$ in terms of the real parts of the eigenvalues of $E$ and the Hausdorff dimension of $B$.