Hamiltonian Systems with Lévy Noise: Symplecticity, Hamilton's Principle and Averaging Principle

Abstract

This work focuses on topics related to Hamiltonian stochastic differential equations with Lévy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of Hamilton's principle by the corresponding formulation of the stochastic action integral and the Euler-Lagrange equation. Based on these properties, we further investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with Lévy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero. Furthermore, we obtain the estimation for the rate of this convergence. Finally, we present an example to illustrate these results.