Observables for Brownian motion on manifolds

TL;DR

This paper investigates how geometry influences Brownian motion on curved manifolds by defining and analyzing various displacement observables, deriving exact formulas, and exploring applications to biological membranes.

Contribution

It introduces new geometric observables for Brownian motion on curved manifolds and derives exact formulas for their expectation values, including in spheres and minimal hypersurfaces.

Findings

Exact formulas for displacement expectations on spheres and minimal hypersurfaces.

Short-time behavior of mean-square Euclidean displacement derived using Weingarten-Gauss equations.

Relevance to lateral diffusion in biological membranes.

Abstract

We study the geometrical influence on the Brownian motion over curved manifolds. We focus on the following intriguing question: what observables are appropriated to measure Brownian motion in curved manifolds? In particular, for those d-dimensional manifolds embedded in $\mathbb{R}^{d+1}$ we define three quantities for the displacement's notion, namely, the geodesic displacement, $s$, the Euclidean displacement, $\deltaR$, and the projected Euclidean displacement $\deltaR_{\perp}$. In addition, we exploit the Weingarten-Gauss equations in order to calculate the mean-square Euclidean displacement's in the short-time regime. Besides, it is possible to prove exact formulas for these expectation values, at all times, in spheres and minimal hypersurfaces. In the latter case, Brownian motion corresponds to the typical diffusion in flat geometries, albeit minimal hypersurfaces are not intrinsically flat. Finally, the two-dimensional case is emphasized since its relation to the lateral diffusion in biological membranes.