The Annals of Applied Probability

Evolutionary Formalism for Products of Positive Random Matrices

Ludwig Arnold, Volker Matthias Gundlach, and Lloyd Demetrius

Full-text: Open access


We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter, we establish a random version of the Perron-Frobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics.

Article information

Ann. Appl. Probab., Volume 4, Number 3 (1994), 859-901.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 28D99: None of the above, but in this section
Secondary: 58F11 92D15: Problems related to evolution 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 92D25: Population dynamics (general)

Evolutionary theory random dynamical system products of random matrices Perron-Frobenius theory Markov chain in a random environment thermodynamic formalism Gibbs measures variational principle equilibrium states


Arnold, Ludwig; Gundlach, Volker Matthias; Demetrius, Lloyd. Evolutionary Formalism for Products of Positive Random Matrices. Ann. Appl. Probab. 4 (1994), no. 3, 859--901. doi:10.1214/aoap/1177004975.

Export citation