Ergodicity of the Fisher infinitesimal model with quadratic selection

TL;DR

This paper proves that in a discrete-time genetic model with quadratic selection, the phenotypic distribution converges exponentially to a unique equilibrium, demonstrating ergodicity and the rapid loss of initial conditions.

Contribution

It extends previous continuous-time results to a discrete setting with quadratic selection, establishing uniqueness and exponential convergence using a global approach.

Findings

Proves exponential convergence to equilibrium distribution.

Establishes ergodicity and rapid forgetting of initial data.

Provides quantitative rates of convergence.

Abstract

We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogues, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.