Projections of spherical Brownian motion

TL;DR

This paper derives a stochastic differential equation for the first coordinates of spherical Brownian motion, analyzing existence and uniqueness despite non-Lipschitz coefficients, and links the radial component to Wright-Fisher diffusions.

Contribution

It provides a new SDE characterization of spherical Brownian motion and its radial component, including a geometric realization of Wright-Fisher diffusions with general parameters.

Findings

Established existence and pathwise uniqueness of the SDE solutions.

Connected the radial component to Wright-Fisher diffusion with mutation.

Provided a geometric realization of Wright-Fisher diffusion as a radial component.

Abstract

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. 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 $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.