Optimal Transport and Convexity in Orlicz Spaces

Abstract

Optimal transport theory has traditionally been developed in the context of classical spaces, where Wasserstein metrics provide a natural geometry on the space of probability measures with finite -moments. However, in many applications-such as in finance, statistical mechanics, and the modelling of heavy-tailed phenomena-classical spaces prove insufficient. This paper proposes a novel generalization of optimal transport to Orlicz spaces, which naturally extend the framework by accommodating more general growth conditions via Young functions. We introduce and rigorously define the Wasserstein-Orlicz metric and explore its topological and geometric properties on spaces of probability measures with finite Orlicz norm. Key results include the extension of Kantorovich duality to the Orlicz setting, as well as the derivation of convexity and lower semicontinuity properties of entropytype functionals under the Wasserstein-Orlicz geometry. Applications are presented in the context of stochastic processes with heavy-tailed distributions, where the classical assumptions of finite p-moments are violated. The proposed framework opens a new avenue for analysing optimal transport problems in nonstandard settings and provides tools for applications in areas where data exhibits non-Gaussian behaviour.