Convergence to the time average by stochastic regularization

TL;DR

This paper compares how quickly a function's average value converges under deterministic Hamiltonian flow versus stochastic perturbations, providing detailed estimates and convergence results in various norms.

Contribution

It offers new quantitative estimates on the convergence rates to the time average under stochastic regularization of Hamiltonian flows, including Sobolev norm convergence.

Findings

Stochastic perturbations can accelerate convergence to the time average.

Explicit estimates are provided in Fourier and Sobolev norms.

Convergence persists even as stochastic perturbation vanishes.

Abstract

We compare the rate of convergence to the time average of a function over an integrable Hamiltonian flow with the one obtained by a stochastic perturbation of the same flow. Precisely, we provide detailed estimates in different Fourier norms and we prove the convergence even in a Sobolev norm for a special vanishing limit of the stochastic perturbation.