Diffusion limit for a slow-fast standard map

TL;DR

This paper proves a central limit theorem for a slow variable in a conjugated standard map with a large parameter, revealing statistical properties and decay of correlations for the system.

Contribution

It introduces a new limit theorem for the standard map with a large parameter, using conjugation and standard pairs techniques.

Findings

Central limit theorem for the slow variable z

Finite-time decay of correlations for the standard map

Related limit theorem for the Chirikov standard map

Abstract

Consider the map $(x, y) \mapsto (x + \epsilon^{-\alpha} \sin (2\pi x) + \epsilon^{-1-\alpha}z, z + \epsilon \sin(2\pi x))$, which is conjugate to the Chirikov standard map with a large parameter. The parameter value $\alpha = 1$ is related to "scattering by resonance" phenomena. For suitable $\alpha$, we obtain a central limit theorem for the slow variable $z$ for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a "finite-time" decay of correlations result.