Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms
Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms
复制标题
微分同胚随机流的李亚普诺夫指数和相对熵
DOI：
10.1007/bf00367301
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发表时间：
1989
影响因子：
2
通讯作者：
P. Baxendale
中科院分区：
文献类型：
--
作者：
P. Baxendale
摘要 d 维流形 M（具有强循环单点运动）的微分同胚随机流的李亚普诺夫指数 λ1≧λ2≧...≧λd 描述了该流下切向量的几乎确定的极限指数增长率。本文展示了 Lyapunov 指数如何与 M 上的随机流和射影丛 PM 上的诱导随机流的测量保留特性相关。相对熵用于量化测量在流动下无法保持不变的程度。结果包括以下内容。如果 M 是紧致的，并且 M 上的单点运动是具有平稳概率测度 ϱ 的非简并扩散，则 λ1+...+λd≤0 相等当且仅当流几乎肯定保持 ϱ 时；另外，如果 PM 上诱发的单点运动满足弱非简并条件，则 λ1=...=λd 当且仅当 M 上存在光滑黎曼结构，相对于该结构，流动几乎肯定是共角的。
SummaryThe Lyapunov exponents λ1≧λ2≧...≧λd for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting exponential growth rates of tangent vectors under the flow. This paper shows how the Lyapunov exponents are related to measure preserving properties of the stochastic flow on M and of the induced stochastic flow on the projective bundle PM. Relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow. The results include the following. If M is compact and if the one-point motion on M is a non-degenerate diffusion with stationary probability measure ϱ then λ1+...+λd≦0 with equality if and only if the flow preserves ϱ almost surely; if in addition the induced one-point motion on PM satisfies a weak non-degeneracy condition then λ1=...=λd if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely.