Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes

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 $d\ge 2$, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension $d\ge 2$ 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 Lévy 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.