Right inverses of Levy processes: the excursion measure in the general case

TL;DR

This paper advances the fluctuation theory of Levy processes by providing an explicit, unified description of the excursion measure away from the minimal right inverse, including new cases with zero Gaussian coefficient.

Contribution

It introduces a comprehensive description of the excursion measure for Levy processes' right inverses, unifying known cases and extending to previously excluded scenarios.

Findings

Unified formula for excursion measure away from the right inverse.

Explicit construction of the Laplace exponent of the minimal right inverse.

Extension to unbounded variation cases with zero Gaussian coefficient.

Abstract

This article is about right inverses of Levy processes as first introduced by Evans in the symmetric case and later studied systematically by the present authors and their co-authors. Here we add to the existing fluctuation theory an explicit description of the excursion measure away from the (minimal) right inverse. This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to excursions away from a point starting negative continuously, and excursions started by a jump, the present description is in terms of excursions away from the supremum continued up to a return time. In the unbounded variation case with zero Gaussian coefficient previously excluded, excursions start negative continuously, but the excursion measures away from the right inverse and away from a point are mutually singular. We also provide a new construction and a new formula for the Laplace exponent of the minimal right inverse.