The Existence of Isometric Stochastic Flows for Riemannian Brownian Motions

Let A be a G-invariant differential operator on a homogeneous space M = G/H, which is the generator of some diffusion process. We study the existence of a G-valued stochastic flow whose one point motion is an A- diffusion in terms of the Lie algebra of G. When M is a Riemannian symmetric space, we show that there exists an isometric stochastic flow whose one point motion is a Brownian motion if M is a symmetric space of compact type and such a flow does not exist if M is of non-compact type. The uniqueness of such a flow is also discussed.