Random Iteration of Maps on a Cylinder and diffusive behavior

Abstract

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: $(\theta ,r)\in \mathbb{T}\times \mathbb{R}=\mathbb{A}$ 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 $\theta$ such that $\int v_{\pm }(\theta ,r)d\theta =0$ for each $r$. We study the random compositions $$({\theta }_n,r_n)=f_{{\omega }_{n-1}}\circ \cdots \circ f_{{\omega }_0}({\theta }_0,r_0)$$ with ${\omega }_k\in \{-1,1\}$ with equal probabilities. We show that under non-degeneracy hypothesis for $n\sim {\epsilon }^{-2}$ 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 $\int v_{\pm }(\theta )d\theta =0$ we prove a vertical central limit theorem. Namely, for $n\sim {\epsilon }^{-2}$ the distributions of $r_n-r_0$ weakly converge to a normal distribution $\mathcal{N}(0,{\sigma }^2)$ for ${\sigma }^2=\frac{1}{4}\int (v_{+}(\theta )-v_{-}(\theta ){)}^{2}d\theta$. 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.