Distance-like Functions On The Wasserstein Space

Optimal transport theory provides a framework for studying the evolution of prob-ability distributions,which has significant relations to Riemann geometry and partial differential equations.In recent years,it has triggered new applications in various fields such as fluid mechanics,urban network and machine learning.The metric geometry of Wasserstein spaces play a key role and is the geometric basis of optimal transport.This paper consider the Wasserstein space P_p(X)on a non-compact,locally compact Polish length space X,i.e.the set of Borel probability measures on the ambient space X with finite p-moments endowed with the Wasserstein distance.Different from Rie-mann manifold and Hadamard space,it is neither locally compact space nor has a proper differential structure,which makes it difficult for some conventional research methods to apply.While only a metric structure is required in the definition of distance-like function,which can replace the viscous solution to the eikonal equation to a certain extent.Two kinds of them are involved in this paper including Busemann functions and horofunctions determined by atom sequences.By means of the theory of displace-ment interpolation of optimal transport,rays in P_p(X)can be described as probability measures concentrated on set of rays in the ambient space.In the conventional theory of global geometry,distance-like functions are closely related to co-rays which are the limits of sequences of specific geodesics.To overcome the non-locally compactness of the space,a new approach is used in this paper to prove that co-rays are always ex-ist for any prescribed initial point in P_p(X).A sequence converges with respect to the Wasserstein distance,if and only if weak convergence and uniformly integrability hold simultaneously.Firstly,weak convergence is obtained by means of the tightness crite-rion on continuous curve space and Prokhorov's theorem.On the other hand,according to the length relation,samples in X are seperated into several cases and estimated re-spectively,thus the whole uniformly integrability is obtained.