Diffusion limit for a slow-fast 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.

23 pages