# Random Iteration of Maps on a Cylinder and diffusive behavior

Research paper by **O. Castejón, V. Kaloshin**

Indexed on: **23 Jan '15**Published on: **23 Jan '15**Published in: **Mathematics - Dynamical Systems**

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#### 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.