%0 Generic %1 Faure2011 %A Faure, Mathieu %A Schreiber, Sebastian J. %D 2011 %K dynamical_systems quasistationarity stochastic_perturbation %T Quasi-Stationary distributions for randomly perturbed dynamical systems %U http://arxiv.org/abs/1101.3420 %X We analyze quasi-stationary distributions $\mu^\epsilon$ of a family of Markov chains $\X^\epsilon\_\epsilon>0$ that are random perturbations of a bounded, continuous map $F:MM$ where $M$ is a subset of $\R^k$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_0M$ such that $F(M_0)=M_0$ and $F(MM_0)=MM_0$. Under some large deviations assumptions on the random perturbations, we show that if there exists a positive attractor for $F$ (i.e. an attractor for $F$ in $MM_0$), then the weak* limit points of $\mu_\epsilon$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of mutation-selection balance, nonequilibrium population dynamics, and evolutionary games.

@misc{Faure2011, abstract = { We analyze quasi-stationary distributions $\mu^\epsilon$ of a family of Markov chains $\{X^\epsilon\}_{\epsilon>0}$ that are random perturbations of a bounded, continuous map $F:M\to M$ where $M$ is a subset of $\R^k$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_0\subset M$ such that $F(M_0)=M_0$ and $F(M\setminus M_0)=M\setminus M_0$. Under some large deviations assumptions on the random perturbations, we show that if there exists a positive attractor for $F$ (i.e. an attractor for $F$ in $M\setminus M_0$), then the weak* limit points of $\mu_\epsilon$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of mutation-selection balance, nonequilibrium population dynamics, and evolutionary games. }, added-at = {2011-02-18T19:19:42.000+0100}, author = {Faure, Mathieu and Schreiber, Sebastian J.}, biburl = {https://www.bibsonomy.org/bibtex/271d4018b40e1f9a7412b6dfe7d75ee8e/peter.ralph}, description = {[1101.3420] Quasi-Stationary distributions for Randomly perturbed dynamical systems}, interhash = {0d57b7197821304ecb5a71b993be3934}, intrahash = {71d4018b40e1f9a7412b6dfe7d75ee8e}, keywords = {dynamical_systems quasistationarity stochastic_perturbation}, note = {cite arxiv:1101.3420 Comment: 31 pages}, timestamp = {2014-11-14T01:48:20.000+0100}, title = {Quasi-Stationary distributions for randomly perturbed dynamical systems}, url = {http://arxiv.org/abs/1101.3420}, year = 2011 }