Stable processes conditioned to hit an interval continuously from the outside

TL;DR

This paper develops a method to condition stable Lévy processes to hit a compact interval continuously from outside, expanding understanding of their boundary behaviors using new harmonic functions.

Contribution

It introduces new harmonic functions and explains how to condition stable processes to hit an interval continuously from outside, advancing the theory of self-similar Markov processes.

Findings

Derived explicit harmonic functions for stable processes

Established a method for continuous hitting conditioning

Extended the understanding of boundary behaviors in stable processes

Abstract

Conditioning stable L\'evy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.