Siegel Brownian motion

TL;DR

This paper introduces Siegel Brownian motion, an analogue of Dyson Brownian motion in the Siegel half-space, describing eigenvalue dynamics driven by stochastic flows and geometric properties like entropy and mean curvature.

Contribution

It constructs Siegel Brownian motion, linking stochastic eigenvalue evolution with geometric and entropy concepts in the Siegel half-space.

Findings

Eigenvalues follow an Ito differential equation related to stochastic gradient ascent.

The entropy function S is the log volume of isospectral orbits and relates to Boltzmann entropy.

In the limit eta= finity, orbits evolve by motion related to mean curvature.

Abstract

We construct an analogue of Dyson Brownian motion in the Siegel half-space H that we term Siegel Brownian motion. Given \beta in (0,\infty], a stochastic flow for Z_t in H is introduced so that the law of the eigenvalues \lambda_t of the cross ratio matrix R(Z_t,iI_n) is determined by the Ito differential equation corresponds to stochastic gradient ascent of a function S. S turns out to be the log volume of isospectral orbit in H and can be understood as a Boltzmann entropy. In the limit \beta=\infty, the group orbits evolve by motion by minus a half times mean curvature.