Stochastic averaging on infinite time interval for a class of nonlinear systems with stochastic perturbation

We investigate stochastic averaging on infinite time interval for a class of continuous-time nonlinear systems with stochastic perturbation and remove several restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, and compactness of the state space of the perturbation process. If an equilibrium of the average system is exponentially stable, we show that the original system is exponentially practically stable in probability. If, in addition, the original system has the same equilibrium as the average system, then the equilibrium of the original system is locally asymptotically stable in probability. These results extend the deterministic general averaging to the stochastic case.