Random Iteration of Cylinder Maps and diffusive behavior away from resonances

Abstract

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: let $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \[ f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) \longmapsto \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r) \\ r+\varepsilon v_{\pm 1}(θ,r) \end{array}\right), \] where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions \[ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0), \] where $ω_k =\pm 1$ with equal probability. We show that under non-degeneracy hypotheses and away from resonances for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a stochastic diffusion process with explicitly computable drift and variance. In the case $u_\pm(θ)=v_\pm(θ)$ are trigonometric polynomials of zero average we prove a vertical central limit theorem, namely, for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to the normal distribution $\mathcal N(0,σ^2)$ with $σ^2=\frac14\int (v_+(θ)-v_-(θ))^2\,dθ$.} The considered random model up to higher order terms in $\varepsilon$ is conjugate to a restrictions to a Normally Hyperbolic Invariant Lamination of the generalized Arnold example. Combining the result of this paper with [8,23,28] we show formation of stochastic diffusive behaviour for the generalized Arnold example.