arXiv subjects

Fabio Cavalletti

Publications and source records attributed to Fabio Cavalletti.

40 records · Page 3Linked to original sources

arXiv2013

Local curvature-dimension condition implies measure-contraction property

We prove that for non-branching metric measure spaces the local curvature condition CDloc(K,N) implies the global version of MCP(K,N). The curvature condition CD(K,N) introduced by the second author and also studied by Lott & Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. It is worth pointing out that our result implies sharp Bishop-Gromov volume growth inequality and sharp Poincaré inequality.

math.MG↗

arXiv2012

Optimal transport with branching distance costs and the obstacle problem

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce two assumptions on the transference plan π which imply that the conditional probabilities of the first marginal on each family of geodesics are continuous and that each family of geodesics is a hourglass-like set. We show that this regularity is sufficient for the construction of a transport map. We apply these results to the Monge problem in d with smooth, convex and compact obstacle obtaining the existence of an optimal map provided the first marginal is absolutely continuous w.r.t. the d-dimensional Lebesgue measure.

math.PR↗

arXiv2011

The Monge problem in Wiener Space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

math.PR↗

arXiv2011

The Monge Problem for distance cost in geodesic spaces

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.

math.PR↗