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.
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