An averaging principle for a completely integrable stochastic Hamiltonian system

TL;DR

This paper proves an averaging principle for small transversal perturbations of integrable stochastic Hamiltonian systems, showing convergence of the action component to a deterministic or stochastic limit as perturbation size diminishes.

Contribution

It establishes an averaging principle for a class of stochastic Hamiltonian systems with explicit convergence rates and extends results to stochastic limits in special cases.

Findings

Action component converges to deterministic system as epsilon approaches zero.

In Hamiltonian perturbation case, the scaled action converges to a stochastic differential equation.

Provides explicit estimates for the rate of convergence.

Abstract

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.