A local limit theorem for a transient chaotic walk in a frozen environment

TL;DR

This paper proves a local limit theorem for a chaotic, deterministic particle walk in a complex environment, explaining why its probability distribution differs from a standard Gaussian at fine scales.

Contribution

It introduces a local limit theorem for a transient chaotic walk in a frozen environment, revealing detailed behavior of the probability distribution at small scales.

Findings

The walk's probability mass function does not converge to a Gaussian density at fine scales.

The limiting distribution over a coarser scale is Gaussian.

Provides detailed explanation of the non-Gaussian local behavior.

Abstract

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk's probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian.