Attraction to and repulsion from a subset of the unit sphere for isotropic stable Lévy processes

Abstract

Taking account of recent developments in the representation of d-dimensional isotropic stable Lévy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that S is a region of the unit sphere Sd−1={x∈Rd:|x|=1}. We construct the aforesaid stable Lévy process conditioned to approach S continuously from either inside or outside of the sphere. Additionally, we show that 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 Döring and Weissman (2020), where similar conditioning is considered, albeit in one dimension as well as providing analogues of the same classical results for Brownian motion, cf. Doob (1957). As in Döring and Weissman (2020), we appeal to recent fluctuation identities related to the deep factorisation of stable processes, cf. Kyprianou (2016), Kyprianou et al. (2020) and Kyprianou et al. (2017).