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

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