$\alpha$-stable L\'evy processes entering the half space or a slab

TL;DR

This paper introduces a new coordinate-based decomposition for isotropic alpha-stable Lévy processes, enabling the derivation of entrance laws into half-spaces and slabs, with numerical methods for first entry laws.

Contribution

It presents a novel orthogonal coordinate decomposition for isotropic alpha-stable Lévy processes and develops new entrance laws, including numerical methods for slabs.

Findings

Derived n-tuple laws for first entrance into half-spaces.

Numerically constructed first entry law into slabs using Monte Carlo.

Enhanced understanding of process behavior in constrained domains.

Abstract

Recent fluctuation identities for $\alpha$-stable L\'evy processes have decomposed paths using generalised spherical polar coordinates revealing an underlying Markov Additive Process (MAP) for which a more advanced form of excursion theory can be exploited. Inspired by this approach, we give a different decomposition of the $d$-dimensional isotropic $\alpha$-stable L\'evy processes in terms of orthogonal coordinates. Accordingly we are able to develop a number of $n$-tuple laws for first entrance into a half-space. We also numerically construct the law of first entry of the process into a slab of the form $(-1, 1)\times \mathbb{R}^{d-1}$ using a walk-on-half-spaces Monte Carlo approach.