Darwinian evolution as Brownian motion on the simplex: A geometric perspective on stochastic replicator dynamics

TL;DR

This paper provides a geometric interpretation of stochastic replicator dynamics as Brownian motion on the simplex, offering new approximation methods and analyzing long-term behavior using advanced mathematical tools.

Contribution

It introduces a geometric perspective on stochastic replicator dynamics via Aitchison geometry and derives new approximation results and long-term behavior analysis.

Findings

Stochastic replicator dynamics can be viewed as Brownian motion on the Aitchison simplex.

Derived approximation results analogous to Wong-Zakai, Donsker's invariance, and JKO schemes.

Analyzed the long-term behavior using Fokker-Planck equations and Wasserstein contraction estimates.

Abstract

We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped the Aitchison geometry. As an immediate consequence we derive three approximation results in the spirit of Wong-Zakai approximation, Donsker's invariance principle and a JKO-scheme. Finally, using the Fokker-Planck equation and Wasserstein-contraction estimates, we study the long time behavior of the stochastic replicator equation, as an example of a non-gradient drift diffusion on the Aitchison simplex.