Attraction to and repulsion from a subset of the unit sphere for isotropic stable L\'evy processes

TL;DR

This paper develops new methods to condition isotropic stable Lévy processes on approaching a subset of the unit sphere from inside or outside, extending one-dimensional results to higher dimensions using fluctuation identities.

Contribution

It introduces novel conditioning techniques for multidimensional stable Lévy processes based on recent self-similar Markov process representations.

Findings

Constructed stable Lévy processes conditioned to approach the sphere from inside or outside.

Extended one-dimensional conditioning results to higher dimensions.

Utilized fluctuation identities related to the deep factorisation of stable processes.

Abstract

Taking account of recent developments in the representation of $d$-dimensional isotropic stable L\'evy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that $\Omega$ is a region of the unit sphere $\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}$. We construct the aforesaid stable L\'evy process conditioned to approach $\mathsf{S}$ continuously from either inside or outside of the sphere. Additionally, we show that %this these processes are in duality with the stable process conditioned to remain inside the sphere and absorb continuously at the origin and to remain outside of the sphere, respectively. Our results extend the recent contributions of D\"oring and Weissman (2018),, where similar conditioning is considered, albeit in one dimension. As is the case there, we appeal to recent fluctuation identities related to the deep factorisation of stable processes.