Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable L\'evy processes

TL;DR

This paper constructs the law of absorption for isotropic stable Lévy processes conditioned to approach specific subsets of the sphere or hyperplane, extending previous work on attraction and repulsion phenomena.

Contribution

It introduces a new framework for understanding how isotropic stable Lévy processes interact with subsets of spheres and hyperplanes, including their duality properties.

Findings

Constructed absorption laws for processes approaching subsets of the sphere and hyperplane.

Established duality between the conditioned process and the underlying Lévy process.

Extended previous results on attraction and repulsion to more general subsets.

Abstract

Suppose that $\mathsf{S}$ is a closed set of the unit sphere $\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}$ in dimension $d\geq2$, which has positive surface measure. We construct the law of absorption of an isotropic stable L\'evy process in dimension $d\geq2$ conditioned to approach $\mathsf{S}$ continuously, allowing for the interior and exterior of $\mathbb{S}^{d-1}$ to be visited infinitely often. Additionally, we show that this process is in duality with the underlying stable L\'evy process. We can replicate the aforementioned results by similar ones in the setting that $\mathsf{S}$ is replaced by $\mathsf{D}$, a closed bounded subset of the hyperplane $\{x\in\mathbb{R}^d : (x, v) = 0\}$ with positive surface measure, where $v$ is the unit orthogonal vector and where $(\cdot,\cdot )$ is the usual Euclidean inner product. Our results complement similar results of the authors Kyprianou, Palau and Saizmaa (2020) in which the stable process was further constrained to attract to and repel from $\mathsf{S}$ from either the exterior or the interior of the unit sphere.