Brownian motion meets Riemann curvature

TL;DR

This paper investigates how Riemannian curvature influences Brownian motion on curved manifolds, deriving a formula for the mean-square geodesic distance and analyzing effects of positive and negative curvature.

Contribution

It introduces a general formula for MSD in curved spaces using Riemann normal coordinates, linking diffusion behavior to curvature invariants.

Findings

Diffusion slows down on positively curved manifolds.

Diffusion speeds up on negatively curved manifolds.

Results are validated on the n-dimensional sphere.

Abstract

The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing on brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so called Riemann normal coordinates to derive a general formula for the mean-square geodesic distance (MSD) at the short-time regime. This formula is written in terms of $O(d)$ invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for the single particle diffusion on biomembranes.