1,721,136 research outputs found
Existence, Uniqueness, and Regularity of Optimal Transport Maps
Full text linkAdapting some techniques and ideas of McCann [Duke Math. J., 80 (1995), pp. 309–323], we extend a recent result with Fathi [Optimal Transportation on Manifolds, preprint] to yield existence and uniqueness of a unique transport map in very general situations, without any integrability assumption on the cost function. In particular this result applies for the optimal transportation problem on an n-dimensional noncompact manifold M with a cost function induced by a C2-Lagrangian, provided that the source measure vanishes on sets with σ-finite (n − 1)-dimensional Hausdorff measure. Moreover we prove that in the case c(x, y) = d2(x, y), the transport map is approximatively differentiable a.e. with respect to the volume measure, and we extend some results of [D. Cordero-Erasquin, R. J. McCann, and M. Schmuckenschlager, Invent. Math., 146 (2001),
pp. 219–257] about concavity estimates and displacement convexity
The Monge problem on non-compact manifolds
No full textIn this paper we prove the existence of an optimal transport map on noncompact manifolds for a large class of cost functions that includes the case c(x, y) = d(x, y), under the only hypothesis that the source measure is absolutely continuous with respect to the volume measure. In particular, we assume compactness neither of the support of the source measure nor of that of the target measure
Rigidity and stability of Caffarelli's log-concave perturbation theorem
No full textIn this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincaré constant as the Gaussian measure, then it almost splits off a Gaussian factor
A mass transportation approach to quantitative isoperimetric inequalities
No full textA sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary
Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields
Full text linkIn this paper we provide a complete analogy between the Cauchy-Lipschitz and
the DiPerna-Lions theories for ODE's, by developing a local version of the
DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a
maximal regular flow for the DiPerna-Lions theory using only local regularity
and summability assumptions on the vector field, in analogy with the classical
theory, which uses only local regularity assumptions. We also study the
behaviour of the ODE trajectories before the maximal existence time. Unlike the
Cauchy-Lipschitz theory, this behaviour crucially depends on the nature of the
bounds imposed on the spatial divergence of the vector field. In particular, a
global assumption on the divergence is needed to obtain a proper blow-up of the
trajectories.In this paper we provide a complete analogy between the Cauchy–Lipschitz and the DiPerna–Lions theories for ODE’s, by developing a local version of the DiPerna–Lions theory. More precisely, we prove the existence and uniqueness of a maximal regular flow for the DiPerna–Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy–Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories
Global sensitivity analysis via optimal transport
No full textWe examine the construction of variable importance measures for multivariate responses using the theory of optimal transport. We start with the classical optimal transport formulation. We show that the resulting sensitivity indices are well-defined under input dependence, are equal to zero under statistical independence, and are maximal under fully functional dependence. Also, they satisfy a continuity property for information refinements. We show that the new indices encompass Wagner's variance-based sensitivity measures. Moreover, they provide deeper insights into the effect of an input's uncertainty, quantifying its impact on the output mean, variance, and higher-order moments. We then consider the entropic formulation of the optimal transport problem and show that the resulting global sensitivity measures satisfy the same properties, with the exception that, under statistical independence, they are minimal but not necessarily equal to zero. We prove the consistency of a given-data estimation strategy and test the feasibility of algorithmic implementations based on alternative optimal transport solvers. Application to the assemble-to-order simulator reveals a significant difference in the key drivers of uncertainty between the case in which the quantity of interest is profit (univariate) or inventory (multivariate). The new importance measures contribute to meeting the increasing demand for methods that make black-box models more transparent to analysts and decision-makers
Corrigendum: Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data
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