Random Iteration of Maps on a Cylinder and diffusive behavior

TL;DR

This paper models random compositions of cylinder maps and demonstrates that, under certain conditions, the distribution of the radial component converges to a diffusion process, revealing insights into stochastic behavior in Hamiltonian systems.

Contribution

It introduces a new probabilistic model of cylinder maps and proves diffusion and normal limit theorems for the radial component under random iteration.

Findings

Radial component converges to a diffusion process with explicit drift and variance.

Vertical CLT shows convergence to a normal distribution with computed variance.

Model relates to stochasticity in nearly integrable Hamiltonian systems.

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}\theta\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}\theta+r+\varepsilon u_{\pm 1}(\theta,r). \\ r+\varepsilon v_{\pm 1}(\theta,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 $$ (\theta_n,r_n)=f_{\omega_{n-1}}\circ \dots \circ f_{\omega_0}(\theta_0,r_0) $$ with $\omega_k \in \{-1,1\}$ with equal probabilities. We show that under non-degeneracy hypothesis for $n\sim \varepsilon^{-2}$ 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}\theta\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}\theta+r+\varepsilon v_{\pm 1}(\theta). \\ r+\varepsilon v_{\pm 1}(\theta) \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 $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a normal distribution $\mathcal N(0,\sigma^2)$ for $\sigma^2=\frac14\int (v_+(\theta)-v_-(\theta))^2\,d\theta$. 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.