This talk will discuss the stochastic area process and its asymptotics on partial flag manifolds with blocks of equal size. A Brownian motion on these manifolds can be represented as a matrix valued diffusion obtained from a unitary Brownian motion in block form. To obtain an explicit expression for the characteristic function of the area processes, the Hermitian Jacobi operators on the simplex are introduced and studied. The eigenpolynomials of these operators simultaneously generalise the Hermitian Jacobi polynomials and the Jacobi polynomials on the simplex. With the help of the stochastic area process, we are able to obtain explicit expressions for different horizontal Brownian motions. This work generalises the results for the full flag manifold, obtained by Fabrice Baudoin, Nizar Demni, Jing Wang and myself, to a more general class of partial flag manifolds.
