Ching-Peng Huang: Matrix Brownian motion and mean curvature flow via Riemannian submersion

Abstract

Using the projection formula for the Beltrami-Laplace operator under Riemannian submersion, we write down matrix Brownian motion on several matrix manifolds of interests and their eigenvalue processes, including Dyson's Brownian motion from the geometric point of view. The mean curvature plays an important role in these analysis. Inspired by this, we discuss ideas of designing stochastic algorithm using the mean curvature flow.