Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds

TL;DR

This paper characterizes invariant manifolds for stochastic differential equations embedded in Euclidean spaces, derives explicit SDEs for Brownian motions on manifolds, and proposes numerical schemes for simulation.

Contribution

It provides explicit formulas for Brownian motions on embedded manifolds and introduces new numerical methods for simulating SDEs on these manifolds.

Findings

Explicit SDEs for Brownian motions on manifolds

Numerical schemes including retraction-based Euler-Maruyama

Simulation results confirming convergence to uniform distributions

Abstract

We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed. We verify numerically that on several compact Riemannian manifolds, the long-term limit of Brownian simulation converges to the uniform distributions, suggesting a method to sample Riemannian uniform distributions