This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein–Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalizing corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin’s seminal work “Sufficient statistics and extreme points” in Ann. Probab. 1978, which contains a purely measure theoretic generalization of the classical analytic Krein–Milman and Choquet Theorems. As an application, we obtain an easy uniqueness proof for T-periodic entrance laws in the general periodic case. A number of further applications to concrete cases are presented.

中文翻译：

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希尔伯特空间中具有 Lévy 噪声的时间非齐次 Ornstein-Uhlenbeck 过程的入口律结构

本文是关于希尔伯特状态空间中具有 Lévy 噪声的时间非齐次 Ornstein-Uhlenbeck 过程的所有入口定律（在 Dynkin 意义上）的结构。我们通过傅里叶变换的明确公式识别具有有限弱一阶矩的极值入口定律，并推广 Dynkin 对维纳噪声和核状态空间的相应结果。然后，我们证明具有有限弱一阶矩的任意入口定律可以唯一地表示为极值上的积分。证明这可以从 Dynkin 的开创性著作《充分统计和极值点》中得到安。概率。1978，其中包含经典分析 Krein-Milman 和 Choquet 定理的纯测度理论推广。作为一个应用程序，我们获得了一个简单的唯一性证明吨- 一般周期情况下的周期进入定律。提出了一些对具体案例的进一步应用。