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On an irreducibility type condition for the ergodicity of nonconservative semigroups
[Sur une condition de type irréductibilité pour l’ergodicité des semi-groupes non conservatifs]
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 733-742.

Nous proposons une condition simple, inspirée des notions d’irréductibilité et d’apériodicité pour les chaînes de Markov, qui permet d’assurer la convergence exponentielle de semi-groupes positifs généraux. Lorsque celle-ci ne s’applique pas sur tout l’espace, elle peut être localisée via l’utilisation de fonctions de Lyapunov. Elle diffère des généralisations habituelles de l’irréductibilité et est basée sur la notion d’accessibilité des trajectoires sous-jacentes. Finalement, cette condition nous permet d’obtenir de nouveaux résultats d’existence d’éléments propres, et les bornes de convergence exponentielle associées, pour un modèle de sélection-mutation en environnement changeant.

We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semigroups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.

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DOI : 10.5802/crmath.92
Classification : 47A35, 35B40, 47D06, 60J80, 92D15, 92D25

Bertrand Cloez 1 ; Pierre Gabriel 2

1 MISTEA, INRAE, Montpellier SupAgro, Univ. Montpellier, 2 place Pierre Viala, 34060 Montpellier, France
2 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     author = {Bertrand Cloez and Pierre Gabriel},
     title = {On an irreducibility type condition for the ergodicity of nonconservative semigroups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {733--742},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.92},
     language = {en},
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Bertrand Cloez; Pierre Gabriel. On an irreducibility type condition for the ergodicity of nonconservative semigroups. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 733-742. doi : 10.5802/crmath.92. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.92/

[1] M. Alfaro; Pierre Gabriel; O. Kavian Confining integro-differential equations from evolutionary biology: ground states and long time dynamics (in preparation)

[2] Vincent Bansaye; Bertrand Cloez; Pierre Gabriel Ergodic behavior of non-conservative semigroups via generalized Doeblin’s conditions, Acta Appl. Math., Volume 166 (2020) no. 1, pp. 29-72 | DOI | MR | Zbl

[3] Vincent Bansaye; Bertrand Cloez; Pierre Gabriel; Aline Marguet A non-conservative Harris ergodic theorem (2019) (https://arxiv.org/abs/1903.03946)

[4] Henri Berestycki; Odo Diekmann; Cornelis J. Nagelkerke; Paul A. Zegeling Can a species keep pace with a shifting climate?, Bull. Math. Biol., Volume 71 (2009) no. 2, pp. 399-429 | DOI | MR | Zbl

[5] Étienne Bernard; Marie Doumic; Pierre Gabriel Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinet. Relat. Models, Volume 12 (2019) no. 3, pp. 551-571 | DOI | MR | Zbl

[6] Reinhard Bürger Perturbations of positive semigroups and applications to population genetics, Math. Z., Volume 197 (1988) no. 2, pp. 259-272 | DOI | MR | Zbl

[7] Reinhard Bürger; Immanuel M. Bomze Stationary distributions under mutation-selection balance: structure and properties, Adv. Appl. Probab., Volume 28 (1996) no. 1, pp. 227-251 | DOI | MR | Zbl

[8] Nicolas Champagnat; Denis Villemonais Exponential convergence to quasi-stationary distribution and Q-process, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 243-283 | DOI | MR | Zbl

[9] Nicolas Champagnat; Denis Villemonais General criteria for the study of quasi-stationarity (2017) (https://arxiv.org/abs/1712.08092)

[10] Jérôme Coville On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equations, Volume 249 (2010) no. 11, pp. 2921-2953 | DOI | MR | Zbl

[11] Jérôme Coville Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., Volume 26 (2013) no. 8, pp. 831-835 | DOI | MR | Zbl

[12] Jérôme Coville; François Hamel On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., Theory Methods Appl., Volume 193 (2019), 111569, 20 pages | Zbl

[13] Pierre Gabriel; Hugo Martin Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation (2019) (https://arxiv.org/abs/1909.08276)

[14] Fang Li; Jerome Coville; Xuefeng Wang On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., Volume 37 (2017) no. 2, pp. 879-903 | MR | Zbl

[15] One-parameter semigroups of positive operators (Rainer J. Nagel, ed.), Lecture Notes in Mathematics, 1184, Springer, 1986 | MR | Zbl

[16] James R. Norris Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, 1998 (reprint of 1997 original) | Zbl

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