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

arXiv: https://arxiv.org/abs/2406.02879 · HTML: https://mlsanity.com/paper/2406.02879/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 45 references that are in the map.
- First posted on arXiv 5 Jun 2024.

## Builds on (most cited first, 5 of 5)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Differential geometry and stochastic dynamics with deep learning numerics](https://mlsanity.com/paper/1712.08364/) (2017)
- [Diffusion Means and Heat Kernel on Manifolds](https://mlsanity.com/paper/2103.00588/) (2021)
- [Rayleigh Quotient Iteration, cubic convergence, and second covariant derivative](https://mlsanity.com/paper/1908.00639/) (2019)
- [Diffusion Means in Geometric Spaces](https://mlsanity.com/paper/2105.12061/) (2021)

## Abstract (arXiv)

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