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 $BC_r$. 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.