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.
Description
[1101.3420] Quasi-Stationary distributions for Randomly perturbed dynamical systems
%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
}