ADInternational audienceWe analyze quasi-stationary distributions \μ ε \ ε\textgreater0 of a family of Markov chains \X ε \ ε\textgreater0 that are random perturbations of a bounded, continuous map F:M→M , where M is a closed subset of R k . Consistent with many models in biology, these Markov chains have a closed absorbing set M 0 ⊂M such that F(M 0 )=M 0 and F(M∖M 0 )=M∖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∖M 0 ), then the weak* limit points of μ ε 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 metapopulations, competing species, host-parasitoid interactions and evolutionary games

We discuss consistency of Vanishingly Smooth Fictitious Play, a strategy in the context of game theory, which can be regarded as a smooth fictitious play procedure, where the smoothing parameter is time-dependent and asymptotically vanishes. This answers a question initially raised by Drew Fudenberg and Satoru Takahashi.

A successful method to describe the asymptotic behavior of various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes is to relate them to an appropriately chosen limit semiflow. Benaïm and Schreiber (2000) define a general class of such stochastic processes, which they call weak asymptotic pseudotrajectories and study their ergodic behavior. In particular, they prove that the weak * limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic semiflow. Continuing a program started by Benaïm, Hofbauer and Sorin (2005), we generalize the ergodic results mentioned above to weak asymptotic pseudotrajectories relative to set-valued dynamical systems.

This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the non convergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of [Hirsch, 1985]. In particular, this allows us to extend significantly the main result of [Hofbauer and Sandholm, 2002] on the convergence of stochastic fictitious play in supermodular games.