On almost surely bounded semigroups of random linear operators

Abstract

Menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry. Following Menger's idea, the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations. The main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators, which is not involved in the classical case. Subsequently, some random operators equations are studied, in particular, we discuss the Schrödinger-type random equation, which considerably generalizes the corresponding result in the sense of Skorohod.