Deep factorisation of the stable process III: Radial excursion theory and the point of closest reach

TL;DR

This paper advances the understanding of isotropic stable processes in multiple dimensions by introducing a new radial excursion theory, analyzing the point of closest reach, and deriving classical identities in this context.

Contribution

It develops a novel radial excursion theory for stable processes, explicitly characterizes the stationary distribution under reflection, and provides a new representation of the Wiener-Hopf factorisation.

Findings

Distribution of the point of closest reach derived

Classical Blumenthal-Getoor-Ray identities extended to n-tuple laws

Explicit stationary distribution for the reflected stable process

Abstract

In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of the recent papers of Kyprianou (2016) and Kyprianou et al. (2017). In particular, we turn our attention to the case of $d$-dimensional isotropic stable process, for $d\geq 2$. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf. Blumenthal et al. (1961), to the setting of $n$-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener-Hopf factorisation of the MAP that underlies the stable process through the Lamperti-Kiu transform.