Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

TL;DR

This paper investigates the long-term behavior of measure-valued birth and death processes with complex dynamics, establishing conditions for extinction, recurrence, and convergence to Yaglom limits across different regimes.

Contribution

It introduces new conditions on Feynman-Kac semigroups to analyze the asymptotics and limits of measure-valued birth and death processes, including the existence of Q-processes and Yaglom limits.

Findings

Proved recurrence for moments and extinction probabilities.

Established convergence in law conditioned on non-extinction.

Identified regimes where Yaglom limits exist in infinite-dimensional settings.

Abstract

In this article we study the long time behaviour of measure-valued birth and death processes in continuous time, where the dynamics between jumps are one-dimensional Markov processes including diffusion and jumps. We consider the three regimes, critical, subcritical and supercritical. Under suitable hypotheses on the Feynman-Kac semigroup, we prove a new recurrence for the moments and the extinction probability, their time asymptotics and the convergence in law for the measure-valued birth and death process conditioned to non extinction, leading to the existence of Q-process and Yaglom limit (in this infinite dimensional setting). We develop three classes of natural examples where our results apply.