Stochastic flows on the boundaries of \(\text{SL}(n,\mathbb{R})\) (Q1326359)

We study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in \(\text{SL}(n,\mathbb{R})\) or \(\text{GL}(n,\mathbb{R})\). These compact spaces are called boundaries of \(\text{SL}(n,\mathbb{R})\), which include \(\text{SO}(n)\), 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 open set whose complement has zero Lebesgue measure. The distance between any two points in the same component of this set tends to zero exponentially 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.