Energy Transport in Random Perturbations of Mechanical Systems

TL;DR

This paper investigates how energy can be transported in a mechanical system with a pendulum and rotator under random Gaussian perturbations, revealing the existence of orbits with energy changes proportional to the perturbation scale.

Contribution

It introduces a novel analysis of energy transport in stochastic Hamiltonian systems, demonstrating the existence of normally hyperbolic invariant manifolds and energy-changing orbits under random perturbations.

Findings

Existence of a distinguished set of times with hyperbolic invariant manifolds.

Orbits where energy changes proportionally to the perturbation scale.

Connection to Arnold diffusion in a stochastic setting.

Abstract

We describe a mechanism for transport of energy in a mechanical system consisting of a pendulum and a rotator subject to a random perturbation. The perturbation that we consider is the product of a Hamiltonian vector field and a scalar, continuous, stationary Gaussian process with H\"older continuous realizations, scaled by a smallness parameter. We show that for almost every realization of the stochastic process, there is a distinguished set of times for which there exists a random normally hyperbolic invariant manifold with associated stable and unstable manifolds that intersect transversally, for all sufficiently small values of the smallness parameter. We derive the existence of orbits along which the energy changes over time by an amount proportional to the smallness parameter. This result is related to the Arnold diffusion problem for Hamiltonian systems, which we treat here in the random setting.