The geometry of random drift I. Stochastic distance and diffusion

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Abstract
A stochastic distance measure is defined for a general diffusion process on a parameter spaceX.This distance is defined bywhere (gij) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij) is the fundamental tensor of the Riemannian space (X, gij), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift withnalleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher tonalleles. The geometry associated with the diffusion equation for random drift withnalleles is that of apart of an (n− 1)-sphere of radius two. We also show that the diffusion equation for random drift isnotspherical Brownian motion, although it is approximated by it near the centroid of frequency space.

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