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.

中文翻译：

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关于希尔伯特空间中的随机流的一个注记

本文是Pesin熵公式和SRB测度的已知结果的扩展，用于将无穷维映射的随机成分映射到随机流的连续时间设置。考虑一个随机流φ上可分离的无限维Hilbert空间保持一个概率测度μ，其被支撑在一个随机紧集ķ。我们发现，如果φ是Ç ²（上ķ）和差的满足一些轻微积的条件，然后Pesin的熵公式成立当μ沿不稳定歧管绝对连续有条件的措施。在K的附加条件下反之亦成立 当系统没有零Lyapunov指数时。