Hausdorff dimension of the range and the graph of stable-like processes

TL;DR

This paper calculates the Hausdorff dimension of the range and graph of stable-like processes, revealing it as a random quantity influenced by the process trajectories, using SDE representations.

Contribution

It introduces a method to determine the Hausdorff dimension of stable-like processes' range and graph via SDE analysis, a novel approach in this context.

Findings

Hausdorff dimension of the process range is random and trajectory-dependent

Method applies to compute the Hausdorff dimension of the graph of these processes

Provides a new framework for analyzing fractal dimensions of jump processes

Abstract

We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE representation of these processes. The method developed here also allows to compute the Hausdorff dimension for the graph.