The Brownian motion and the canonical stochastic flow on a symmetric space

Liao, Ming

doi:10.1090/S0002-9947-1994-1129436-2

The Brownian motion and the canonical stochastic flow on a symmetric space
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Trans. Amer. Math. Soc. 341 (1994), 253-274 Request permission

Abstract:

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 canonical stochastic flow ${F_t}$ on $O(V)$. We show that almost surely, ${x_t}$ has a limiting direction as it goes to infinity. The study of the asymptotic stability of ${F_t}$ is reduced to the study of the limiting behavior of the adjoint action on the Lie algebra $\mathcal {G}$ of $G$ by the horizontal diffusion in $G$. We determine the Lyapunov exponents and the associated filtration of ${F_t}$ in terms of root space structure of $\mathcal {G}$.

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