Evolutionary formalism for products of positive random matrices
type: Item
instance of: scholarly article
title: Evolutionary formalism for products of positive random matrices
zbMATH Open document ID: 0818.15015
published in: The Annals of Applied Probability
publication date: 1994-11-22T00:00:00Z
review text: It is proved that for a product of (componentwise) positive random matrices there exists a unique positive random unit vector \(u\) whose span coincides with the Oseledets space associated with the largest Lyapunov exponent. Further, \(u\) is Markov (i.e., measurable with respect to the past), and all Oseledets spaces associated with smaller Lyapunov exponents are orthogonal to \(u\). This `random Perron-Frobenius theorem' generalizes several results previously obtained by others. The theorem is applied to develop a thermodynamic formalism, and it is shown that the top Lyapunov exponent is an analytic function of the generator of the product of random matrices. The results are then used to discuss the dynamic behaviour of replicating entities -- molecules, cells, or higher organisms.
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 92D15
zbMATH DE Number: 681406
zbMATH Keywords: product of random matrices
zbMATH Keywords: random Perron-Frobenius theorem
zbMATH Keywords: Oseledets space
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: dynamic behaviour of replicating entities
zbMATH Keywords: molecules
zbMATH Keywords: cells
author: Ludwig Arnold
author: Matthias Gundlach
author: Lloyd Demetrius
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1177004975
OpenAlex ID: W1990055914
DOI: 10.1214/AOAP/1177004975
description: scientific article; zbMATH DE number 681406