Supperdiffusions for certain nonuniformly hyperbolic systems

TL;DR

This paper studies superdiffusion phenomena in nonuniformly hyperbolic systems, providing explicit formulas for superdiffusion constants and demonstrating the convergence to normal distribution under abnormal central limit theorem conditions.

Contribution

It introduces a martingale approximation approach and derives explicit formulas for superdiffusion constants in various nonuniformly hyperbolic models.

Findings

Superdiffusion characterized by non-linear mean square displacement growth.

Explicit formula for superdiffusion constant related to system structure.

Applicable to models like Lorentz gas, billiards with cusps, and more.

Abstract

We investigate superdiffusion for stochastic processes generated by nonuniformly hyperbolic system models, in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We construct a martingale approximation that follows the idea of Doob's decomposition theorem. We obtain an explicity formula for the superdiffusion constant in terms of the fine structure that originates in the phase transitions as well as the geometry of the configuration domains of the systems. Models that satisfy our main assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to other nonuniformly hyperbolic systems with slow correlation decay rates of order $\cO(1/n)$.