Summary
Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal Δ in M 2 of the associated two point motion. Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M 2-Δ if the top Lyapunov exponent is positive.
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At the period when this research was initiated, both authors where guests of the I.M.A. in Minneapolis. The first author was at Aberdeen University, Scotland when this article was prepared. Throughout the period of this research, the second author has been partially supported by N.S.F. grant DMS-8611487 and ARO grant DAAL03-86-K-171
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Baxendale, P.H., Stroock, D.W. Large deviations and stochastic flows of diffeomorphisms. Probab. Th. Rel. Fields 80, 169–215 (1988). https://doi.org/10.1007/BF00356102
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DOI: https://doi.org/10.1007/BF00356102