Some applications of duality for L\'evy processes in a half-line

TL;DR

This paper establishes an analytic duality relation for Lévy processes killed upon exiting a half-line, leading to new time-reversal identities and applications to self-similar Markov processes and spatial stationarity.

Contribution

It introduces a novel duality relation for Lévy processes in a half-line and applies it to construct spatially stationary Lévy processes and derive Lamperti-type representations.

Findings

Derived a duality relation for killed Lévy processes.

Constructed a spatially stationary Lévy process indexed by the real line.

Provided a Lamperti-type representation for self-similar Markov processes.

Abstract

The central result of this paper is an analytic duality relation for real-valued L\'evy processes killed upon exiting a half-line. By Nagasawa's theorem, this yields a remarkable time-reversal identity involving the L\'evy process conditioned to stay positive. As examples of applications, we construct a version of the L\'evy process indexed by the entire real line and started from $-\infty$ which enjoys a natural spatial-stationarity property, and point out that the latter leads to a natural Lamperti-type representation for self-similar Markov processes in $(0,\infty)$ started from the entrance point 0+.