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

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.