  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.
