Summary
The Lyapunov exponents λ 1≧λ 2≧...≧λ d 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 λ 1+...+λ d ≦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 λ 1=...=λ d if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely.
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Baxendale, P.H. Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Th. Rel. Fields 81, 521–554 (1989). https://doi.org/10.1007/BF00367301
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DOI: https://doi.org/10.1007/BF00367301
Keywords
- Entropy
- Manifold
- Probability Measure
- Exponential Growth
- Statistical Theory