Stable L\'evy processes in a cone

TL;DR

This paper explores the behavior of stable Lévy processes in cones, developing new notions of conditioning and analyzing their properties as self-similar Markov processes, with implications for understanding their structure and long-term behavior.

Contribution

It introduces the concepts of stable processes conditioned to remain in or absorb at a cone's apex, and analyzes their properties via Lamperti-Kiu decomposition, addressing open questions about Markov additive processes.

Findings

Constructed recurrent extensions of stable processes in cones

Established duality between two conditioning types

Provided conditions for stationary distributions of ladder MAPs

Abstract

Ba\~nuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (L\'evy) process in dimension $d\geq 2$ exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution? We show how the two forms of conditioning are dual to one another. Moreover, we construct the recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes. In the spirit of several very recent works, the results presented here show that many previously unknown results of stable processes, which have long since been understood for Brownian motion, or are easily proved for Brownian motion, become accessible by appealing to the notion of the stable process as a self-similar Markov process, in addition to its special status as a L\'evy processes with a semi-tractable potential analysis.