Hofbauer, Josef et A. Imhof, Lorens, Time averages, recurrence and transience in the stochastic replicator dynamics, Annals of applied probability , 19(4), 2009, pp. 1347-1368
We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.