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.

Symplectic Euler scheme for Hamiltonian stochastic differential equations driven by Levy noise

“…where X = (P, Q), X 0 = (p, q) with X 0 < +∞ and V r = (− ∂Hr ∂Q , ∂Hr ∂P ), r = 0, 1, 2, ..., m. Here the norm • is defined as (4), and P, Q, p, q are n-dimensional column-vectors. We assume that the functions V r , r = 0, 1, 2, ..., m, satisfy the conditions in [13] such that Hamiltonian SDEs (3) have a unique global solution, and the solution process is adapted and càdlàg.…”

“…Many useful contributions are made in developing special numerical methods and the corresponding numerical analysis of SDEs [9], [11,12], [14]- [17]. Second, the construction of conditions which can preserve the Hamiltonian structure of SDEs driven by non-Gaussian noise has been presented in [13]. These results are the foundations of symplectic scheme of Hamiltonian SDEs in the sense of Marcus form.…”

Homogenization of Dissipative Hamiltonian Systems under Lévy Fluctuations

“…Hamiltonian dynamics [24], as an equivalent description of Newton's second law in the framework of classical mechanics, form the framework of statistical mechanics. Dissipative Hamiltonian systems with noise have been investigated recently [25,26].…”

Formulation of Stochastic Contact Hamiltonian Systems

“…A stochastic Hamiltonian system preserving symplectic structure was proposed by Bismut 16 . Following this framework, Li 17 and Wei 18 discussed averaging principles for completely integrable stochastic Hamiltonian systems. Moreover, the symplectic structure-preserving numerical algorithms are developed.…”