An averaging principle for a completely integrable stochastic Hamiltonian system

We investigate the effective behaviour of a small transversal perturbation of order to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions H_i, i = 1, ..., n. An averaging principle is shown to hold 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 perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at 1/² converges to that of a limiting stochastic differentiable equation.