Uniform contractivity of the Fisher infinitesimal model with strongly convex selection

Abstract

The Fisher infinitesimal model is a classical model of phenotypic trait inheritance in quantitative genetics. Here, we prove that it encompasses a remarkable convexity structure which is compatible with a selection function having a convex shape. It yields uniform contractivity along the flow, as measured by a $L^{\infty }$ version of the Fisher information. It induces in turn asynchronous exponential growth of solutions, associated with a well-defined, log-concave, equilibrium distribution. Although the equation is non-linear and non-conservative, our result shares some similarities with the Bakry-Emery approach to the exponential convergence of solutions to the Fokker-Planck equation with a convex potential. Indeed, the contraction takes place at the level of the Fisher information. Moreover, the key lemma for proving contraction involves the Wasserstein distance $W_{\infty }$ between two probability distributions of a (dual) backward-in-time process, and it is inspired by a maximum principle by Caffarelli for the Monge-Ampère equation.

28 pages, 2 figures

References in corpus (2)

• The Cauchy problem for the infinitesimal model in the regime of small variance
• Ergodicity of the Fisher infinitesimal model with quadratic selection

Cited by in corpus (1)

• On the Fisher infinitesimal model without variability