In this note we prove that in a metric measure space (X,d,m) verifying the measure contraction property with parameters K∈R and 1<∞, 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.
Optimal maps in essentially non-branching spaces / F. Cavalletti, A. Mondino. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - 19:6(2017 Dec), pp. 1750007.1-1750007.27. [10.1142/S0219199717500079]
Optimal maps in essentially non-branching spaces
F. Cavalletti
Primo
;
2017
Abstract
In this note we prove that in a metric measure space (X,d,m) verifying the measure contraction property with parameters K∈R and 1<∞, 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.
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