A "Gaussian" for diffusion on the sphere

TL;DR

This paper introduces an analytical approximate propagator for diffusion on the sphere, analogous to the Gaussian for planar diffusion, which is effective for both short and intermediate times and enables efficient numerical simulations.

Contribution

It provides a nonperturbative, closed-form approximation for spherical diffusion, extending short-time heat kernel methods and improving numerical integration efficiency.

Findings

The new formula accurately approximates diffusion on the sphere for various times.

Monte Carlo simulations show the new algorithm outperforms previous methods.

The approach facilitates studying large deviations in spherical diffusion.

Abstract

We present an analytical closed form expression, which gives a good approximate propagator for diffusion on the sphere. Our formula is the spherical counterpart of the Gaussian propagator for diffusion on the plane. While the analytical formula is derived using saddle point methods for short times, it works well even for intermediate times. Our formula goes beyond conventional "short time heat kernel expansions" in that it is nonperturbative in the spatial coordinate, a feature that is ideal for studying large deviations. Our work suggests a new and efficient algorithm for numerical integration of the diffusion equation on a sphere. We perform Monte Carlo simulations to compare the numerical efficiency of the new algorithm with the older Gaussian one.