[Sur une condition de type irréductibilité pour l’ergodicité des semi-groupes non conservatifs]
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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@article{CRMATH_2020__358_6_733_0,
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},
}
TY - JOUR AU - Bertrand Cloez AU - Pierre Gabriel TI - On an irreducibility type condition for the ergodicity of nonconservative semigroups JO - Comptes Rendus. Mathématique PY - 2020 SP - 733 EP - 742 VL - 358 IS - 6 PB - Académie des sciences, Paris DO - 10.5802/crmath.92 LA - en ID - CRMATH_2020__358_6_733_0 ER -
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/
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