A Non-Conservative Harris' Ergodic Theorem

Vincent Bansaye (1) , Bertrand Cloez (2) , Pierre Gabriel (3) , Aline Marguet (4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
3 LMV - Laboratoire de Mathématiques de Versailles
4 IBIS - Modeling, simulation, measurement, and control of bacterial regulatory networks

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 spectral gaps, complementing Krein-Rutman theorems and generalizing recent results relying on probabilistic approaches. The proof is based on a non-homogenous h-transform of the semi-group and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris' theorem for conservative semigroups. We apply these results and 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.

Mots clés: branching processes; Positive semigroups; ergodicity; measure solutions; growth-fragmentation; Krein-Rutman theorem; quasi-stationary distribution

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Vincent Bansaye, Bertrand Cloez, Pierre Gabriel, Aline Marguet. A Non-Conservative Harris' Ergodic Theorem. 2019. ⟨hal-02062882v1⟩