An averaging principle for integrable stochastic Hamiltonian systems

Consider a stochastic differential equation whose diffusion vector fields are formed from an integrable family of Hamiltonian functions Hi, i = 1, . . . n. We investigate the effect of a small transversal perturbation of order to such a system. An averaging principle is shown to hold for this system and the action component of the solution converges, as ! 0, to the solution of a deterministic system of differential equations when the time is rescaled at 1/ . An estimate for the rate of the convergence is given. In the case when the limiting deterministic system is constant we show that the action component of the solution scaled at 1/ 2 converges to that of a limiting stochastic differentiable equation.