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Pesin’s Formula for Random Dynamical Systems on \(\mathbf {R}^d\)

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Abstract

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on \(\mathbf {R}^d\) which have an invariant probability measure absolutely continuous to the Lebesgue measure on \(\mathbf {R}^d\). Finally we will show that a broad class of stochastic flows on \(\mathbf {R}^{d}\) of a Kunita type satisfies Pesin’s formula.

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Acknowledgments

The present research was supported by the International Research Training Group Stochastic Models of Complex Processes funded by the German Research Council (DFG). The author gratefully thanks Michael Scheutzow and Simon Wasserroth from TU Berlin for their support and several fruitful discussions.

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  1. Institut für Mathematik, MA 7-4, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Moritz Biskamp

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  1. Moritz Biskamp
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Biskamp, M. Pesin’s Formula for Random Dynamical Systems on \(\mathbf {R}^d\) . J Dyn Diff Equat 26, 109–142 (2014). https://doi.org/10.1007/s10884-014-9347-4

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  • DOI: https://doi.org/10.1007/s10884-014-9347-4

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