Abstract
This paper is an extension of known results of Pesin’s entropy formula and SRB measures for random compositions of infinite-dimensional mappings to the continuous-time setting of stochastic flows. Consider a stochastic flow ϕ on a separable infinite dimensional Hilbert space preserving a probability measure μ, which is supported on a random compact set K. We show that if ϕ is C2 (on K) and satisfies some mild integrable conditions of the differentials, then Pesin’s entropy formula holds if μ has absolutely continuous conditional measures along the unstable manifolds. The converse is also true under an additional condition on K when the system has no zero Lyapunov exponent.
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Acknowledgments
The last author would also like to thank Professor Jon Aaronson and the School of mathematical sciences of Tel Aviv University for hospitality during his visit there.
Funding
This work was supported by National Natural Science Foundation of China (No.11871394), Natural Science Foundation of Shaanxi Province (No. 2019JM-123), and Israel Science Foundation (No.1289/17).
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Tang, D., Gu, L. & Li, Z. A Remark on Stochastic Flows in a Hilbert Space. J Dyn Control Syst 26, 775–783 (2020). https://doi.org/10.1007/s10883-020-09481-7
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DOI: https://doi.org/10.1007/s10883-020-09481-7