[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.
Accepté le :
Accepté après révision le :
Publié le :
@article{CRMATH_2020__358_6_733_0,
author = {Cloez, Bertrand and Gabriel, Pierre},
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},
url = {http://www.numdam.org/articles/10.5802/crmath.92/}
}
TY - JOUR AU - Cloez, Bertrand AU - Gabriel, Pierre 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 UR - http://www.numdam.org/articles/10.5802/crmath.92/ DO - 10.5802/crmath.92 LA - en ID - CRMATH_2020__358_6_733_0 ER -
%0 Journal Article %A Cloez, Bertrand %A Gabriel, Pierre %T On an irreducibility type condition for the ergodicity of nonconservative semigroups %J Comptes Rendus. Mathématique %D 2020 %P 733-742 %V 358 %N 6 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.92/ %R 10.5802/crmath.92 %G en %F CRMATH_2020__358_6_733_0
Cloez, Bertrand; Gabriel, Pierre. On an irreducibility type condition for the ergodicity of nonconservative semigroups. Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 733-742. doi : 10.5802/crmath.92. http://www.numdam.org/articles/10.5802/crmath.92/
[1] Confining integro-differential equations from evolutionary biology: ground states and long time dynamics (in preparation)
[2] 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] A non-conservative Harris ergodic theorem (2019) (https://arxiv.org/abs/1903.03946)
[4] Can a species keep pace with a shifting climate?, Bull. Math. Biol., Volume 71 (2009) no. 2, pp. 399-429 | DOI | MR | Zbl
[5] 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] Perturbations of positive semigroups and applications to population genetics, Math. Z., Volume 197 (1988) no. 2, pp. 259-272 | DOI | MR | Zbl
[7] Stationary distributions under mutation-selection balance: structure and properties, Adv. Appl. Probab., Volume 28 (1996) no. 1, pp. 227-251 | DOI | MR | Zbl
[8] Exponential convergence to quasi-stationary distribution and -process, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 243-283 | DOI | MR | Zbl
[9] General criteria for the study of quasi-stationarity (2017) (https://arxiv.org/abs/1712.08092)
[10] 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] Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., Volume 26 (2013) no. 8, pp. 831-835 | DOI | MR | Zbl
[12] On generalized principal eigenvalues of nonlocal operators with a drift, Nonlinear Anal., Theory Methods Appl., Volume 193 (2019), 111569, 20 pages | Zbl
[13] Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation (2019) (https://arxiv.org/abs/1909.08276)
[14] 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 (Nagel, Rainer J., ed.), Lecture Notes in Mathematics, 1184, Springer, 1986 | MR | Zbl
[16] Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, 1998 (reprint of 1997 original) | Zbl
Cité par Sources :
