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Characterization of absolutely continuous curves in Wasserstein spaces

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Abstract

Let X be a separable, complete metric space and \(\fancyscript{P}_{p}(X)\) be the space of Borel probability measures with finite moment of order p > 1, metrized by the Wasserstein distance. In this paper we prove that every absolutely continuous curve with finite p-energy in the space \(\fancyscript{P}_{p}(X)\) can be represented by a Borel probability measure on C([0,T];X) concentrated on the set of absolutely continuous curves with finite p-energy in X. Moreover this measure satisfies a suitable property of minimality which entails an important relation on the energy of the curves. We apply this result to the geodesics of \(\fancyscript{P}_{p}(X)\) and to the continuity equation in Banach spaces.

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  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100, Pavia, Italy

    Stefano Lisini

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  1. Stefano Lisini
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Correspondence to Stefano Lisini.

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Communicated by L. Ambrosio

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Lisini, S. Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. 28, 85–120 (2007). https://doi.org/10.1007/s00526-006-0032-2

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