A model of invariant control system using mean curvature drift from Brownian motion under submersions

TL;DR

This paper constructs a stochastic process on Riemannian manifolds whose image follows a mean curvature flow of fibers, enabling explicit Brownian motion models on spaces like positive definite matrices and exploring invariant control systems.

Contribution

It introduces a novel model linking Brownian motion with mean curvature flow via Riemannian submersions, providing explicit formulas and applications to homogeneous spaces.

Findings

Explicit formulas for mean curvature and drift in matrix form

Brownian motions on spaces like Poincaré upper half plane and positive definite matrices

Eigenvalue processes resemble Dyson's Brownian motion

Abstract

Given a submersion $\phi: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=\phi(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $\pi: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincar\'e's upper half plane and the Bures-Wasserstein geometry on $\pdef$, on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian motion. By choosing the background metric via natural $GL(n)$ action, we arrive at an invariant control system on the $GL(n)$-homogenous space $GL(n)/O(n)$. We investigate feasibility of developing stochastic algorithms using the mean curvature flow. KEY WORDS: mean curvature flow, gradient flow, Brownian motion, Riemannian submersion, random matrix, eigenvalue processes, geometry of positive definite matrices, stochastic algorithm, control theory on homogeneous space