Skew-product decomposition of Brownian motion on ellipsoid

TL;DR

This paper derives a skew-product decomposition of Brownian motion on a specific ellipsoid, showing that its projection onto the last coordinate behaves like a Wright-Fisher diffusion with an unusual selection coefficient.

Contribution

It introduces a novel decomposition for Brownian motion on certain ellipsoids and links the projection to a Wright-Fisher diffusion with atypical parameters.

Findings

Projection onto the last coordinate is a Wright-Fisher diffusion.

Decomposition provides new insights into Brownian motion on ellipsoids.

Results connect geometric stochastic processes with population genetics models.

Abstract

In this article we obtain a skew-product decomposition of a Brownian motion on an ellipsoid of dimension $n$ in a Euclidean space of dimension $n+1$. We only consider such ellipsoid whose restriction to first $n$ dimensions is a sphere and its last coordinate depends on a variable parameter. We prove that the projection of this Brownian motion on to the last coordinate is, after a suitable transformation, a Wright-Fisher diffusion process with atypical selection coefficient.