  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.
