Open Access
2023 Time reversal of spinal processes for linear and non-linear branching processes near stationarity
Benoît Henry, Sylvie Méléard, Viet Chi Tran
Author Affiliations +
Electron. J. Probab. 28: 1-27 (2023). DOI: 10.1214/23-EJP911


We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times tT. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.

Funding Statement

This work has been supported by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X and by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. V.C.T. also acknowledges support from Labex Bézout (ANR-10-LABX-58).


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Benoît Henry. Sylvie Méléard. Viet Chi Tran. "Time reversal of spinal processes for linear and non-linear branching processes near stationarity." Electron. J. Probab. 28 1 - 27, 2023.


Received: 14 January 2022; Accepted: 30 January 2023; Published: 2023
First available in Project Euclid: 22 February 2023

arXiv: 2201.05223
MathSciNet: MR4551564
Digital Object Identifier: 10.1214/23-EJP911

Primary: 60F99 , 60J80 , 60K35 , 92D15 , 92D25

Keywords: ancestral path , birth-death processes , competition , genealogy , interaction , jump process , Many-to-One formula , non-local mutation operator , phylogeny , stochastic individual-based models

Vol.28 • 2023
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