We consider the problem of minimizing an action potential that arises from large deviation theory for stochastic approximations. The solutions to the minimizing problem satisfy, in the sense of a viscosity solution, a Hamilton-Jacobi equation. From weak KAM theory, we know that these viscosity solutions are characterised by the projected Aubry set. The main result of this paper is that, for a specific rate function corresponding to a stochastic approximation algorithm, we prove that the projected Aubry set is equal to the forward limit set to the limit ODE.
