Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type
Abstract
We study Brownian motion on Hermitian symmetric spaces of non-compact type in their bounded-domain realization. Using Jordan triple systems, we identify the spectral values after an appropriate change of variables as a Heckman-Opdam diffusion of type BCr. We then analyze two Brownian functionals: the symplectic area associated with the canonical Kähler form, and, in the tube-type case, the winding defined by the Jordan determinant. For the area process we prove a martingale representation, a central limit theorem, and an exact conditional characteristic function expressed as a ratio of Heckman-Opdam heat kernels. For the determinant winding process we obtain analogous heat kernel formulas and prove convergence to a Cauchy law with scale determined by the initial determinant. These results extend classical formulas of Paul Lévy and Marc Yor from the Euclidean setting to the full class of Hermitian symmetric spaces of non-compact type.
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BibTeX
@misc{baudoin2026geometric,
title = {{Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type}},
author = {Fabrice Baudoin and Alexandre Reber},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.28377},
eprint = {2605.28377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.28377},
}
RIS
TY - UNPB
TI - Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type
AU - Fabrice Baudoin
AU - Alexandre Reber
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2605.28377
UR - https://arxiv.org/abs/2605.28377
AB - We study Brownian motion on Hermitian symmetric spaces of non-compact type in their bounded-domain realization. Using Jordan triple systems, we identify the spectral values after an appropriate change of variables as a Heckman-Opdam diffusion of type BC_r. We then analyze two Brownian functionals: the symplectic area associated with the canonical Kähler form, and, in the tube-type case, the winding defined by the Jordan determinant. For the area process we prove a martingale representation, a central limit theorem, and an exact conditional characteristic function expressed as a ratio of Heckman-Opdam heat kernels. For the determinant winding process we obtain analogous heat kernel formulas and prove convergence to a Cauchy law with scale determined by the initial determinant. These results extend classical formulas of Paul Lévy and Marc Yor from the Euclidean setting to the full class of Hermitian symmetric spaces of non-compact type.
ER -