On random convex analysis

Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is convex analysis over random locally convex modules. Since random locally convex modules have the more complicated topological and algebraic structures than ordinary locally convex spaces, establishing random convex analysis will encounter harder mathematical challenges than classical convex analysis so that there are still a lot of fundamentally important unsolved problems in random convex analysis. This paper is devoted to solving some important theoretic problems. First, we establish the inferior limit behavior of a proper lower semicontinuous $L^0$--convex function on a random locally convex module endowed with the locally $L^0$--convex topology, which makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we investigate the relations among continuity, locally $L^0$--Lipschitzian continuity and almost surely sequent continuity of a proper $L^0$--convex function. And then, we establish the elegant relationships among subdifferentiability, G\^ateaux--differentiability and Fr\'ech\'et--differentiability for a proper $L^0$--convex function defined on random normed modules. At last, based on the Ekeland's variational principle for a proper lower semicontinuous $\bar{L}^0$--valued function, we show that $\varepsilon$--subdifferentials can be approximated by subdifferentials. We would like to emphasize that the success of this paper lies in simultaneously considering the $(\varepsilon, \lambda)$--topology and the locally $L^0$--convex topology for a random locally convex module.