Diffusion on a Hypersphere: Application to the Wright-Fisher model

TL;DR

This paper develops a mathematical framework connecting diffusion processes on a hypersphere with the Wright-Fisher genetic model, enabling better simulation and interpretation of genetic drift with mutation.

Contribution

It introduces a novel eigenfunction expansion approach that links hyperspherical diffusion to the Wright-Fisher model using Ito calculus, providing new tools for analysis and simulation.

Findings

Eigenfunction expansion offers a simple interpretation of Wright-Fisher dynamics.

The method facilitates efficient simulation of the Wright-Fisher model.

Connection established between hyperspherical diffusion and genetic drift models.

Abstract

The eigenfunction expansion by Gegenbauer polynomials for the diffusion on a hypersphere is transformed into the diffusion for the Wright-Fisher model with a particular mutation rate. We use the Ito calculus considering stochastic differential equations. The expansion gives a simple interpretation of the Griffiths eigenfunction expansion for the Wright-Fisher model. Our representation is useful to simulate the Wright-Fisher model as well as Brownian motion on a hypersphere.