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August, 1994 Evolutionary Formalism for Products of Positive Random Matrices
Ludwig Arnold, Volker Matthias Gundlach, Lloyd Demetrius
Ann. Appl. Probab. 4(3): 859-901 (August, 1994). DOI: 10.1214/aoap/1177004975


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.


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Ludwig Arnold. Volker Matthias Gundlach. Lloyd Demetrius. "Evolutionary Formalism for Products of Positive Random Matrices." Ann. Appl. Probab. 4 (3) 859 - 901, August, 1994.


Published: August, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0818.15015
MathSciNet: MR1284989
Digital Object Identifier: 10.1214/aoap/1177004975

Primary: 28D99
Secondary: 54H20‎ , 58F11 , 60J10 , 92D15 , 92D25

Keywords: equilibrium states , Evolutionary theory , Gibbs measures , Markov chain in a random environment , Perron-Frobenius theory , Products of random matrices , Random dynamical system , Thermodynamic formalism , Variational principle

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 3 • August, 1994
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