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Optimal maps in essentially non-branching spaces

Fabio Cavalletti

Dipartimento di Matematica, Università degli Studi di Pavia, Italy

SISSA, Trieste, Italy

E-mail Address: cavallet@sissa.it

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and

Andrea Mondino

Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

E-mail Address: A.Mondino@warwick.ac.uk

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https://doi.org/10.1142/S0219199717500079Cited by:35 (Source: Crossref)

Abstract

In this paper we prove that in a metric measure space $(X, d, 𝔪)$ verifying the measure contraction property with parameters $K \in ℝ$ and $1 < N < \infty$ , any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $𝔪$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

Keywords:

AMSC: 53C23, 49Q20

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