Hausdorff dimension of the graph of an operator semistable L\'evy process

TL;DR

This paper calculates the Hausdorff dimension of the graph of an operator semistable Lévy process, linking it to the eigenvalues of the process's exponent and the dimension of the time set.

Contribution

It provides a formula for the Hausdorff dimension of the graph of operator semistable Lévy processes based on spectral properties and the dimension of the domain set.

Findings

Hausdorff dimension expressed via eigenvalues of E

Dimension depends on the Hausdorff dimension of B

Results extend understanding of fractal properties of Lévy process graphs

Abstract

Let $X=\{X(t):t\geq0\}$ be an operator semistable L\'evy process in $\mathbb{R}^d$ with exponent $E$, where $E$ is an invertible linear operator on $\mathbb{R}^d$. For an arbitrary Borel set $B\subseteq\mathbb{R}_+$ we interpret the graph $Gr_X(B)=\{(t,X(t)):t\in B\}$ as a semi-selfsimilar process on $\mathbb{R}^{d+1}$, whose distribution is not full, and calculate the Hausdorff dimension of $Gr_X(B)$ in terms of the real parts of the eigenvalues of the exponent $E$ and the Hausdorff dimension of $B$.