随机局部凸模上L0-值函数次微分和近似次微分性质

 研究了随机凸分析中的次微分与近似次微分问题。在层次结构分析以及最近建立的随机局部凸模上的分离定理基础上，本文得到了关于随机局部凸模上L⁰值函数的次微分和近似次微分的一些良好基本性质。首先由随机局部凸模上点与集合的分离定理，证明了：对于具有可数连结性质的随机局部凸模上真的、T_c-下半连续的、L⁰(F)-凸函数，它的近似次微分非空；再者，由随机局部凸模上两个集合的分离定理，证明了对于具有可数连结性质的随机局部凸模上两个真的、T_c-下半连续的、L⁰(F)-凸函数，及其次微分公式。其中，L⁰(F)表示定义在概率空间上的广义实值随机变量等价类全体形成的集合。本文推广了经典凸分析中的相应结果。

Properties of Subdifferentials and Approximate Subdifferentials of L0-Valued Functions on Random Locally Convex Modules

The random convex analysis of subdifferentials and approximate subdifferentials is carried out in this paper. Based on the analysis of the stratification structure and the recently developed separation theorems in random locally convex modules, some nice basic properties about subdifferentials and approximate subdifferentials of a L⁰-valued function on random locally convex modules are obtained. The main results are as follows. First, for a proper and T_c-lower semicontinuous L⁰-valued L⁰(F)-convex function f defined on a random locally convex module (E, Λ) where both E and Λ have the countable concatenation property, it is proven that the approximate subdifferential of f is nonempty. Second, for two propers and T_c-lower semicontinuous L⁰-valued L⁰(F)-convex functions, denoted by f and g, which are defined on a random locally convex module (E, Λ) where both E and Λ have the countable concatenation property, the formula about the subdifferentials of f+g is given.