Random Iteration of Maps on a Cylinder and diffusive behavior

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \begin{eqnarray} \nonumber 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), \end{eqnarray} 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 $(θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0)$ with $ω_k \in \{-1,1\}$ with equal probabilities. We show that under non-degeneracy hypothesis for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a diffusion process with explicitly computable drift and variance. In the case of random iteration of the standard maps \begin{eqnarray} \nonumber f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}θ+r+\varepsilon v_{\pm 1}(θ). \\ r+\varepsilon v_{\pm 1}(θ) \end{array}\right), \end{eqnarray} where $v_\pm$ are trigonometric polynomials such that $\int v_\pm(θ)\,dθ=0$ we prove a vertical central limit theorem. Namely, for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a normal distribution $\mathcal N(0,σ^2)$ for $σ^2=\frac14\int (v_+(θ)-v_-(θ))^2\,dθ$. Such random models arise as a restrictions to a Normally Hyperbolic Invariant Lamination for a Hamiltonian flow of the generalized example of Arnold. We expect that this mechanism of stochasticity sheds some light on formation of diffusive behaviour at resonances of nearly integrable Hamiltonian systems.

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