Stochastic analysis on finite dimensional Siegel disks, approach to the infinite dimensional Siegel disk and upper half-plane

AbstractWe consider the group G of complex matrices of order 2n of the form g=ABBA such that tgJg=J with J=0I−I0. The notation tM for a matrix means the matrix transposed of M. In the first part, for G, we explicit polar coordinates and a Cartan decomposition. We discuss how to define the radial part in the group G. We write in the Hua–Siegel polar coordinates the Stratonovich stochastic system (S)dA=B∘dXanddB=A∘dX where g=ABBA∈G, and where X is a Brownian motion in the vector space of symmetric complex matrices of order n. We obtain in polar coordinates, the stochastic differential equations satisfied by the symmetric matrix Z=BA−1. We also write (S) in Iwasawa coordinates on the group G