Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions

  • Vincent Bansaye
  • Bertrand Cloez
  • Pierre GabrielEmail author


We provide quantitative estimates in total variation distance for positive semigroups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin’s type conditions inherited from Champagnat and Villemonais (Probab. Theory Relat. Fields 164(1–2):243–283, 2016) for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semigroups and new bounds in the homogeneous setting. They are illustrated on the renewal equation.


Positive semigroups Non-autonomous linear evolution equations Measure solutions Ergodicity Krein-Rutman theorem Floquet theory Branching processes Population dynamics 

Mathematics Subject Classification (2010)

35B40 47A35 47D06 60J80 92D25 



B.C. and V.B. have received the support of the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MnHn-FX. The three authors have been supported by ANR projects, funded by the French Ministry of Research: B.C. by ANR PIECE (ANR-12-JS01-0006-01), V.B. by ANR ABIM (ANR-16-CE40-0001) and ANR CADENCE (ANR-16-CE32-0007), and P.G. by ANR KIBORD (ANR-13-BS01-0004).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CMAPÉcole PolytechniquePalaiseau cedexFrance
  2. 2.MISTEA, INRA, Montpellier SupAgroUniv. MontpellierMontpellierFrance
  3. 3.Laboratoire de Mathématiques de Versailles, UVSQ, CNRSUniversité Paris-SaclayVersailles cedexFrance

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