Abstract We investigate an evolutionary process that is continually perturbed by "mutations." If the support of the (unique) stochastically stable distribution is a singleton, then it must be a Nash equilibrium. If one element of a "mutation-connected component" of Nash equilibria appears in the stochastically stable distribution, then all members of that component appear. This implies that the stochastically stable distribution will include weakly dominated strategies in many cases, and in some cases the outcome may include only weakly dominated strategies. Choice trembles can eliminate weakly dominated strategies if but only if they occur with arbitrarily higher probability than that of mutations. Journal of Economic Literature Classification Numbers: C70, C72.