STOCHASTIC FLOWS ON THE BOUNDARIES OF SL(N,R)

We study the asymptotic stability of the stochastic flows on a class o f compact spaces induced by a diffusion process in SL(n, R) or GL(n, R ). These compact spaces are called boundaries of SL(n, R), which inclu de SO(n), the flag manifold, the sphere S(n-1) and the Grassmannians. The one point motions of these flows are Brownian motions. For almost every omega, we determine the set of stable points. This is a random o pen set whose complement has zero Lebesgue measure. The distance betwe en any two points in the same component of this set tends to zero expo nentially fast under the flow. The Lyapunov exponents at stable points are computed explicitly. We apply our results to a stochastic flow on S(n-1) generated by a stochastic differential equation which exhibits some nice symmetry.