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Transactions of the American Mathematical Society

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

Author: Ming Liao
Journal: Trans. Amer. Math. Soc. 341 (1994), 253-274
MSC: Primary 58G32; Secondary 60J65
MathSciNet review: 1129436
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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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