Abstract
We study rays and co-rays in the Wasserstein space \(P_p({\mathcal {X}})\) (\(p > 1\)) whose ambient space \({\mathcal {X}}\) is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space. We show the existence of co-rays for any prescribed initial probability measure. We introduce Busemann functions on the Wasserstein space and show that co-rays are negative gradient lines in some sense.
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The authors thank the anonymous referee for helpful suggestions.
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Communicated by A. Malchiodi.
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The third author is supported by the National Natural Science Foundation of China (Grants 11631006, 11790272, 11571166). The Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.
