随机赋范模的几何理论及其应用

Motivated by the applications of random normed modules to conditional risk measures, conditional optimization and the local theory of Banach spaces, after the work of random strict convexity and random uniform convexity, the project is focused on the further study of geometric theory of random normed modules. First, the notions of Gateaux differentiability and Fréchet differentiability are introduced for L^{0}-valued functions on a random normed module, and smooth and uniformly smooth random normed modules are respectively defined. Next, the dual relations between random smoothness and random strict convexity, random uniform smoothness and random uniform convexity are discussed, respectively. Then, the connections between random smoothness (or random uniform smoothness) of a random normed module and classical smoothness (resp. classical uniform smoothness) of the abstract L^{p}-space derived from the random normed module are studied. Finally, the notion of L^{0}-extreme point is introduced in a random normed module and its relations to ordinary extreme point in the abstract L^{p}-space derived from the random normed module are established in the expectation of their potential applications to the famous open problem "Are Krein-Milman property and Radon-Nikodym property equivalent to each other?".

受随机赋范模在条件风险、条件优化和Banach空间局部理论中的应用启发，继最近关于随机严格凸性和随机一致凸性的工作，本项目对随机赋范模的几何理论展开进一步研究。首先对随机赋范模上的L^{0}-值函数定义Gateaux可微性与Fréchet可微性，给出随机光滑与随机一致光滑的随机赋范模的定义；接着讨论随机光滑性、随机一致光滑性分别与随机严格凸性和随机一致凸性之间的对偶关系；然后，研究随机光滑性、随机一致光滑性分别与随机赋范模生成的抽象L^{p}-空间的经典光滑性和一致光滑性之间的内在联系；最后，对随机赋范模引进L^{0}-端点的概念，建立它与随机赋范模生成的抽象L^{p}-空间中通常端点之间的关系，以期探索它们在著名问题"Krein-Milman性质与Radon-Nikodym性质是否等价"中的应用。

随着随机度量理论的日益发展，涌现出许多新的方法和工具。围绕随机赋范模几何理论的研究，综合运用泛函分析经典理论以及随机赋范模的新工具，得到了下述主要成果：.(1)给出从随机局部凸模到广义实值可测函数等价类的真的局部函数在其有效域内的方向导数、Gateaux可微性及Frechet可微性的定义，进一步揭示了随机赋范模上真的L^0凸函数的次可微性、Gateaux可微性与Frechet可微性之间的巧妙的联系。.(2)建立了有可数连接性质的实L^0准桶模中的真下半连续L^0凸函数的连续性和次可微性定理；.(3)证明了条件风险度量模途径所使用的模型空间L^{p}_{F}(E)是经典赋范空间L^{p}(E)的可数连接包，从而为研究三种条件风险度量之间的联系提供了关键技巧；.(4)经由随机闭单位球的支撑泛函给出随机光滑性的定义，建立了随机光滑性与随机严格凸性之间的联系，给出了随机光滑性的一些等价刻画。.这些结果是随机赋范模几何理论的实质性发展。特别是其中关于可微性的结果也促进了随机凸分析的发展。