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

Abstract

Given a Riemannian submersion ϕ : M → N \phi : M \to N , we construct a stochastic process X X on M M such that the image Y ≔ ϕ ( X ) Y≔\phi (X) is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping π : G L ( n ) → G L ( n ) / O ( n ) \pi : GL(n) \to GL(n)/O(n) , whose image is equivalent to the space of n n -by- n n positive definite matrices, S + ( n , n ) \mathcal {S}_+(n,n) , and the said flow 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 Poincaré’s upper half plane and the Bures-Wasserstein geometry on S + ( n , n ) \mathcal {S}_+(n,n) , on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson’s Brownian motion. By choosing the background metric via natural G L ( n ) GL(n) action, we arrive at an invariant control system on the G L ( n ) GL(n) -homogenous space G L ( n ) / O ( n ) GL(n)/O(n) . We investigate the feasibility of developing stochastic algorithms using the mean curvature flow.