Branching processes seen from their extinction time via path decompositions of reflected L\'evy processes

TL;DR

This paper studies the time-reversal invariance of excursions of spectrally positive Lévy processes, revealing new dualities and invariance properties of branching processes and their genealogical structures at extinction.

Contribution

It proves the invariance of pre- and post-$oldsymbol{ extgamma}$ subpaths of Lévy excursions under space-time reversal, with implications for branching process dualities.

Findings

Pre-$oldsymbol{ extgamma}$ and post-$oldsymbol{ extgamma}$ paths are space-time reversible.

Local time process of the excursion is invariant when viewed backward.

Certain critical branching processes are invariant under time reversal at extinction.

Abstract

We consider a spectrally positive L\'evy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process \cite{GaJa98,Lam10}. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf_{[0,t]} X$ and we let $\gamma$ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma$ and the post-$\gamma$ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of L\'evy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.