We describe the large deviations properties, stationary distribution asymptotics,

and stochastically stable states of stochastic evolutionary processes based on the logit

choice rule, focusing on behavior in the small noise double limit. These aspects of the

stochastic evolutionary process can be characterized in terms of solutions to certain

minimum cost path problems. We solve these problems explicitly using tools from

optimal control theory. The analysis focuses on three-strategy coordination games

that satisfy the marginal bandwagon property and that have an interior equilibrium,

but our approach can be applied to other classes of games and other choice rules.