Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first n coordinates of a Brownian motion on the unit sphere in Rn+ℓ. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in Rn+ℓ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.