The notion of convexity and concavity on Wiener space

TL;DR

This paper extends the concepts of convexity and concavity to the setting of Wiener space, establishing their properties and demonstrating that key inequalities from finite-dimensional analysis also hold in this infinite-dimensional context.

Contribution

It introduces definitions of convexity and concavity for equivalence classes of random variables on Wiener space and proves their relevance to fundamental inequalities.

Findings

Convexity and concavity are defined for random variables on Wiener space.

Finite-dimensional inequalities are generalized to the Wiener space setting.

The work bridges finite and infinite-dimensional analysis through these notions.

Abstract

We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting.