Data di Pubblicazione:
2013
Citazione:
The Monge Problem for Distance Cost in Geodesic Spaces / S. Bianchini, F. Cavalletti. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 318:3(2013 Feb 13), pp. 615-673. [10.1007/s00220-013-1663-8]
Abstract:
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w. r. t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Statistical and Nonlinear Physics; Mathematical Physics;
Elenco autori:
S. Bianchini, F. Cavalletti
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