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Abstract We study the Brownian motion on the non-compact Grassmann manifold $$\frac {\textbf {U}(n-k,k)} {\textbf {U}(n-k)\textbf {U}(k)}$$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $$\textbf {U}(n-k,k)$$ are then given.
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Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions
We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.
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- PAR ID:
- 10221353
- Date Published:
- Journal Name:
- Electronic journal of probability
- Volume:
- 26
- Issue:
- 38
- ISSN:
- 1083-6489
- Page Range / eLocation ID:
- 1-21
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1214/21-EJP600
