Geometric ergodicity of a stochastic Hamiltonian system

TL;DR

This paper proves that a stochastic Hamiltonian system with noise and perturbation has solutions that exponentially converge to a unique invariant measure, despite the system's potential for finite-time explosion.

Contribution

It extends previous results by showing exponential ergodicity of the system under stochastic forcing and perturbation, ensuring long-term statistical stability.

Findings

Solutions are exponentially attracted to a unique invariant measure.

The system exhibits noise-induced stability despite finite-time explosion potential.

The results generalize previous boundedness findings to exponential convergence.

Abstract

We study the long time statistics of a two-dimensional Hamiltonian system in the presence of Gaussian white noise. While the original dynamics is known to exhibit finite time explosion, we demonstrate that under the impact of the stochastic forcing as well as a deterministic perturbation, the solutions are exponentially attractive toward the unique invariant probability measure. This extends previously established results in which the system is shown to be noise-induced stable in the sense that the solutions are bounded in probability.