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 K\"ahler 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 L\'{e}vy and Marc Yor from the Euclidean setting to the full class of Hermitian symmetric spaces of non-compact type.