Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits

Abstract

We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced by isometric action of compact Lie groups, whose orbits have nonzero mean curvature, which contributes to drift terms and is the log gradient of orbit volume function, showing in another way that eigenvalues collide whenever the fibre is degenerate. We thus provide a unified treatment and better connection between eigenvalue processes in different settings with the language of Riemannian geometry. Under such interpretation, we see how we can naturally recover eigenvector processes and derive $\beta$ process such as $\beta$-Dyson Brownian motion for general $\beta>0$. \textbf{KEYWORDS}: eigenvalue process, symmetric space, mean curvature flow, gradient flow, isospectral manifold, random matrix ensemble