Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view

We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small ϵ>0. The o.d.e. limit corresponding to ϵ=0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the ϵ↓0 limit, the invariant density has the form ≈exp(−W(x)/ϵ2), where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given.