An averaging principle for a completely integrable stochastic Hamiltonian system

We investigate the effective behaviour of a small transversal perturbation of order $\epsilon$ 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 $H_i, i=1,\dots n$. An averaging principle is shown to hold and the action component of the solution converges, as $\epsilon \to 0$, to the solution of a deterministic system of differential equations when the time is rescaled at $1/\epsilon$. 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 $1/\epsilon^2$ converges to that of a limiting stochastic differentiable equation.