Evolutionary formalism for products of positive random matrices
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.
dynamic behaviour of replicating entities
scientific article; zbMATH DE number 681406