Diffusion in a weakly random Hamiltonian flow

TL;DR

This paper studies how particles in a weakly random Hamiltonian flow behave like Brownian motion over certain scales, providing error estimates and applying results to acoustic wave energy diffusion.

Contribution

It introduces a rigorous analysis of particle trajectories converging to Brownian motion and extends findings to acoustic wave energy diffusion in random media.

Findings

Particle trajectories converge to spatial Brownian motion.

Error estimates for stochastic acceleration to momentum diffusion.

Energy density of acoustic waves diffuses spatially.

Abstract

We consider the motion of a particle governed by a weakly random Hamiltonian flow. We identify temporal and spatial scales on which the particle trajectory converges to a spatial Brownian motion. The main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentum diffusion. We also apply our results to the system of random geometric acoustics equations and show that the energy density of the acoustic waves undergoes a spatial diffusion.