Stable processes, self-similarity and the unit ball

TL;DR

This paper reviews classical and recent identities related to isotropic stable processes within the unit ball, combining potential analysis and self-similar Markov process theories to deepen understanding and present new proofs.

Contribution

It unifies classical potential analytical identities with modern self-similar Markov process perspectives, offering new proofs and extending existing results.

Findings

Revisits classical identities for isotropic stable processes

Combines potential analysis with self-similar Markov process theory

Provides new proofs and extensions of known results

Abstract

Around the 1960s a celebrated collection of papers emerged offering a number of explicit identities for the class of isotropic stable processes in one and higher dimensions; these include, for example, the lauded works of Blumenthal, Getoor, Ray, Port and Rogozin. Amongst other things, these results nicely exemplify the use of standard Riesz potential theory on the unit open ball $\mathbb{B}_d :=\{ x\in \mathbb{R}^d :|x|<1\}$, $\mathbb{R}^d\backslash\mathbb{B}_d$ and $\mathbb{S}_d :=\partial \mathbb{B}_d$ with the, then, modern theory of potential analysis for Markov processes. Following initial observations of Lamperti in 1972, with the occasional sporadic work of Kiu, Vuolle-Apiala and Graversen in the 1980s, an alternative understanding of stable processes through the theory of self-similar Markov processes has prevailed in the last decade or more. This point of view offers deeper probabilistic insights into some of the aforementioned potential analytical relations. In this review article, we will rediscover many of the aforementioned classical identities in relation to the unit ball and recent extensions thereof by combining elements of these two theories, which have otherwise been largely separated by decades in the literature. We present a dialogue that appeals as much as possible to path decompositions. Most notable in this respect is the Lamperti-Kiu decomposition of self-similar Markov processes and the Riesz-Bogdan-\.Zak transformation. Some of the proofs we give are known, some are known proofs mixed with new methods and some proofs are completely new. We assume that the reader has a degree of familiarity with the bare basics of L\'evy processes but nonetheless, we often include reminders of standard material.