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Viscosity solutions to the eikonal equations play a fundamental role to studythe geometry, topology and geodesic flows. The classical definition ofviscosity solution depends on the differential structure and can not extenddirectly to a general metric space. However, the distance-like functions, whichare exactly viscosity solutions of the eikonal equation on a Riemannianmanifold, is independent of the differential structure and well-defined on anon-compact, complete, separable metric space. In this paper, we study theviscosity solutions of the eikonal equation on Wasserstein space Pp(X) (p > 1),whose ambient space X is complete, separable, non-compact, locally compactlength space. But the Wasserstein space is not locally compact, the co-rays (orcalibrated curves) of the distance-like functions may not exist. Fortunately,we show that if the distance-like function is induced by a sequence of closedsubset diverging to infinity consisting of Dirac probability measures in theWasserstein space, the co-rays do indeed exist. The concrete conditions of theexistence of the co-rays are also provided. We also show that a distance-likefunction on the ambient space can induce a distance-like function on theassociated Wasserstein space.

General distance-like functions on the Wasserstein space