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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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Quaternionic Brownian Windings
We define and study the three-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the asymp- totic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior. The corresponding asymptotic law seems to be new and is related to the Cauchy relativistic distribution.
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- PAR ID:
- 10221362
- Date Published:
- Journal Name:
- Journal of Theoretical Probability
- ISSN:
- 0894-9840
- 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/10.1007/s10959-020-01034-9
