The convergence to equilibrium of neutral genetic models

Pierre del Moral (1, 2) , Laurent Miclo (3) , Frédéric Patras (4) , Sylvain Rubenthaler (4)

1 A3 - Advanced analysis to code optimization
2 IMB - Institut de Mathématiques de Bordeaux
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 JAD - Laboratoire Jean Alexandre Dieudonné

This article is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

Domaines: Probabilités [math.PR]

Pierre del Moral, Laurent Miclo, Frédéric Patras, Sylvain Rubenthaler. The convergence to equilibrium of neutral genetic models. [Research Report] RR-6442, 2008, pp.20. ⟨inria-00239259v2⟩