Probabilistic models of multidimensional piecewise expanding mappings
type: Item
instance of: scholarly article
title: Probabilistic models of multidimensional piecewise expanding mappings
zbMATH Open document ID: 0833.58027
DOI: 10.1007/BF02186286
author: Jörn Loviscach
published in: Journal of Statistical Physics
publication date: 1996-03-25T00:00:00Z
review text: Because strongly chaotic systems exhibit diffusion-like behavior in the sense of central limit theorems, the author of this paper works toward the goal of finding more precise statements about the similarity to probabilistic diffusion by studying how the evolution of probability densities under \(d\)-dimensional piecewise expanding mappings can be modeled by Markov processes with smooth transition probabilities. The author investigates a class of discrete time dynamical systems given by: \[ X_{n+ 1}= X_n+ f(X_n, y_n),\quad y_{n+ 1}= y_n+ g(X_n, y_n), \] where \(X_n\in \mathbb{R}^d\) and \(y_n\) lies on an \(\ell\)-dimensional torus, and where \(f\) and \(g\) are integer-periodic with respect to \(X_n\). One then regards \(X_n\) as an observable, \(y_n\) as a hidden variable, and looks for a probabilistic system which approximates the behavior of \(X_n\) as \(n\to \infty\). To do this, the author fixes probability densities for \(X_0\) and \(y_0\) and treats \(X_n\) as a random variable with density function \(\rho^d_n\). Then he searches for a stochastic process whose long term density \(\rho_n\) matches the long term behavior of \(X_n\) in both microscopic \((\sim 1)\) and macroscopic \((\sim\sqrt n)\) scales. The author establishes that, in the general case, the decay \(|\rho^d_n- \rho_n|\) in the \(L^1(\mathbb{R}^d)\) norm is faster than \(n^{-\gamma}\) for all \(\gamma< 1/2\). This probabilistic approximation is of global and local type.
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 70K50
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 796342
zbMATH Keywords: multidimensional piecewise expanding maps
zbMATH Keywords: deterministic chaos
zbMATH Keywords: central limit theorem
zbMATH Keywords: local limit theorem
zbMATH Keywords: Markov process
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02186286
OpenAlex ID: W1999682356
description: scientific article; zbMATH DE number 796342