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We prove that stochastic replicator dynamics can be interpreted as intrinsicBrownian motion on the simplex equipped the Aitchison geometry. As an immediateconsequence we derive three approximation results in the spirit of Wong-Zakaiapproximation, Donsker's invariance principle and a JKO-scheme. Finally, usingthe Fokker-Planck equation and Wasserstein-contraction estimates, we study thelong time behavior of the stochastic replicator equation, as an example of anon-gradient drift diffusion on the Aitchison simplex.

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