Abstract.
Let φ t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ t = ξ t ψ t such that ξ t is isometric, ψ t fixes m and Dψ t (u) = us t , where s t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ t (u) = u t s t with u t being an orthonormal frame. As an application, we can show that ψt preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates.
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Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000
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Liao, M. Decomposition of stochastic flows and Lyapunov exponents. Probab Theory Relat Fields 117, 589–607 (2000). https://doi.org/10.1007/PL00008736
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DOI: https://doi.org/10.1007/PL00008736