THE BROWNIAN-MOTION AND THE CANONICAL STOCHASTIC FLOW ON A SYMMETRICAL SPACE

We study the limiting behavior of Brownian motion x(t) on a symmetric space V = G/K of noncompact type and the asymptotic stability of the c anonical stochastic flow E(t) on O(V). We show that almost surely, x(t ) has a limiting direction as it goes to infinity. The study of the as ymptotic stability of F-t is reduced to the study of the limiting beha vior of the adjoint action on the Lie algebra G of G by the horizontal diffusion in G. We determine the Lyapunov exponents and the associate d filtration of F-t in terms of root space structure of G.