Absolutely continuous curves in extended Wasserstein-Orlicz spaces

Introduction

In this paper, we extend a previous result of the author [Lis07] to a more general class of spaces. The result in [Lis07] concerns the representation of absolutely continuous curves with finite energy in the Wasserstein space (P(X,d), Wp) (the space of Borel probability measures on a Polish metric space (X,d), endowed with the p-Wasserstein distance induced by d) by means of superposition of curves of the same kind on the space (X,d). The superposition is described by a probability measure on the space of continuous curves in (X,d) representing the curve in (P(X,d), Wp) and satisfying a suitable property.

The first extension of the previous representation result considers an extended Polish space (X, τ,d) instead of a Polish space (X,d). In the second extension, we consider the ψ-Orlicz-Wasserstein distance induced by an increasing convex function ψ: [0,+∞)→[0,+∞] instead of the p-Wasserstein distance modelled on the particular case of ψ(r) =r^p for p >1.

Extended Polish Spaces and Probability Measures

The class of extended Polish spaces was introduced in the recent paper [AGS14]. The authors consider a Polish space (X, τ), i.e. τ is a separable topology on X induced by a distance δ on X such that (X, δ) is complete. The Wasserstein distance is defined between Borel probability measures on (X, τ) and constructed by means of an extended distance d on X that can assume the value +∞. The minimization problem defining the extended Wasserstein distance makes sense between Borel probability measures on (X, τ), assuming that the extended distance d is lower semi-continuous with respect to τ.

Continuous Curves in Extended Wasserstein-Orlicz Spaces

Given ψ satisfying (6), (10) and (11). If u:I→(X,d) is right continuous at every point and continuous except at most a countable set, and (15) lim sup h→0+ ||d(u(·+h), u(·))/h||__{L ψ (I)} < +∞, where u is extended for t > T as u(t) =u(T), then u ∈ AC__ψ (I; (X,d)).

Thus, for every u ∈ AC__ψ (I; (X,d)), there exists the following limit, called metric derivative, (14) |u′|(t) := lim h→0 d(u(t+h), u(t))/|h| for L^1 -a.e.t∈I. The function t → |u′|(t) belongs to L__ψ (I) and it is the minimal one that satisfies (13). We call a curve u ∈ AC__ψ (I; (X,d)) an absolutely continuous curve with finite L__ψ -energy.

Conclusion

A final application of the main theorem is to characterize the geodesics of the Wasserstein-Orlicz space. This important result extends the representation theorem to a wider class of spaces and distances. As future work, it remains to be seen how these results could be applied to other types of spaces or measures.