On the harmonic measure of stable processes

TL;DR

This paper provides explicit formulas for the harmonic measure and Green functions of stable Lévy processes on finite intervals, unifying and extending previous results with new identities and computations.

Contribution

It introduces a unified approach using hypergeometric identities to evaluate harmonic measures and Green functions for stable processes, offering new explicit formulas and insights.

Findings

Explicit harmonic measure formulas for stable processes

Unified proof of existing results in the literature

Computed hitting probabilities and characterized harmonic functions

Abstract

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real L{\'e}vy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the hitting probabilities of points and describe the non-negative harmonic functions for the stable process killed outside a finite interval.