Stable processes with reflection

TL;DR

This paper introduces a new method to construct and analyze a reflected isotropic alpha-stable Lévy process, including its return and stationary distributions, using nonlocal Schrödinger perturbations and a novel ladder process.

Contribution

It develops a new analytic approach for concatenating Markov processes via nonlocal Schrödinger perturbations and introduces a ladder process to handle reflections.

Findings

Constructed a reflected isotropic alpha-stable Lévy process.

Described the return and stationary distributions of the process.

Introduced a novel ladder process encoding reflections.

Abstract

We construct a Hunt process that can be described as an isotropic $\alpha$-stable L\'evy process reflected from the complement of a bounded open Lipschitz set. In fact, we introduce a new analytic method for concatenating Markov processes. It is based on nonlocal Schr\"odinger perturbations of sub-Markovian transition kernels and the construction of two supermedian functions with different growth rates at infinity. We apply this framework to describe the return distribution and the stationary distribution of the process. To handle the strong Markov property at the reflection time, we introduce a novel ladder process, whose transition semigroup encodes not only the position of the process, but also the number of reflections.