# A non-conservative Harris ergodic theorem

## TL;DR

This paper extends Harris's ergodic theorem to non-conservative positive semigroups, providing conditions for exponential convergence and spectral gap estimates, with applications to population dynamics and PDEs.

## Contribution

It introduces a non-conservative Harris-type theorem using an $h$-transform and Lyapunov functions, generalizing classical results to broader semigroup settings.

## Key findings

- Exponential convergence of birth-death processes conditioned on survival
- Spectral gap estimates for growth-fragmentation PDEs
- Existence of Perron eigenelements under new conditions

## Abstract

We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of the spectral gap, complementing Krein-Rutman theorems and generalizing probabilistic approaches. The proof is based on a non-homogenous $h$-transform of the semigroup and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris's theorem for conservative semigroups and recent techniques developed for the study of absorbed Markov processes. We apply these results to population dynamics. We obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs.