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 ) / ϵ 2 ) , 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.