Diffusion limit for a slow-fast standard map

Consider the map $(x, y) \mapsto (x + ε^{-α} \sin (2πx) + ε^{-1-α}z, z + ε\sin(2πx))$, which is conjugate to the Chirikov standard map with a large parameter. The parameter value $α= 1$ is related to "scattering by resonance" phenomena. For suitable $α$, 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.

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