随机动力系统不变测度的Lipschitz正则性
Lipschitz regularity of the invariant measure of random dynamical systems
AI总结：
本文研究随机动力系统不变测度分解的正则性，通过构造类Wasserstein-Kantorovich度量的各向异性赋范空间并分析转移算子动力学，利用泛函分析技术证明谱隙，进而证明Lipschitz可观测量的指数衰减相关性和统计性质。
摘要：
本文推导了与一类随机动力系统（RDS）相关的不变测度分解的正则性结果。通过构造由类Wasserstein-Kantorovich度量定义的合适各向异性赋范空间，并理解相关转移算子在其不动点邻域内的动力学，得到了本文结果。具体而言，我们运用泛函分析技术证明了其在有号测度的合适空间上作用的谱隙。我们应用此分析证明了Lipschitz可观测量的指数衰减相关性和RDS的统计性质。
英文摘要：
In this article we derive a regularity result for the disintegration of the invariant measure associated to a class of Random Dynamical Systems - RDS. The results of this work are obtained by constructing a suitable anisotropic normed space defined by the Wasserstein-Kantorovich-like metric and understanding the dynamics of the associated transfer operator in a neighborhood of its fixed point. Precisely, we employ functional analytic techniques to demonstrate a spectral gap for its action on suitable spaces of signed measures. We apply this analysis to prove an exponential decay of correlation statement for Lipschitz observables and statistical properties of the RDS.
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