Brownian motion and stochastic areas on complex partial flag manifolds with blocks of equal size

TL;DR

This paper constructs a matrix-valued Brownian motion on complex partial flag manifolds, derives explicit characteristic functions for stochastic areas, and reveals their limit distribution as multivariate Cauchy, linking to windings on Stiefel manifolds.

Contribution

It introduces a new family of diffusions on complex flag manifolds, providing explicit formulas for stochastic areas and their distributions, and connects these to winding functionals on Stiefel manifolds.

Findings

Explicit characteristic function for stochastic areas on flag manifolds

Limit law of stochastic areas is a multivariate Cauchy distribution

New diffusions generalizing Jacobi and Hermitian Jacobi processes

Abstract

We construct a Brownian motion on complex partial flag manifolds with blocks of equal size as a matrix-valued diffusion from a Brownian motion on the unitary group. This construction leads to an explicit expression for the characteristic function of the joint distribution of the stochastic areas on these manifolds. The limit law of these stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries. By relating the area functionals on flag manifolds to the winding functional on the complex Stiefel manifold, we establish new results about simultaneous Brownian windings on the complex Stiefel manifold. To establish these results, this work introduces a new family of diffusions, which generalise both the Jacobi processes on the simplex and the Hermitian Jacobi processes.