Kantorovich duality for general transport costs and applications

  title={Kantorovich duality for general transport costs and applications},
  author={Nathael Gozlan and Cyril Roberto and Paul-Marie Samson and Prasad Tetali},
  journal={arXiv: Probability},
Applications of weak transport theory
Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali introduced a transport problem for `weak' cost functionals. Basic results of optimal transport theory can be
Existence, duality, and cyclical monotonicity for weak transport costs
The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on
Characterization of a class of weak transport-entropy inequalities on the line
We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does
On a mixture of Brenier and Strassen Theorems
We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405).
Around the entropic Talagrand inequality
In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of
A new class of costs for optimal transport planning
We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target
On the convex Poincaré inequality and weak transportation inequalities
We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This
A Theory of Transfers: Duality and convolution
We introduce and study the permanence properties of the class of linear transfers between probability measures. This class contains all cost minimizing mass transports, but also martingale mass
Weak Optimal Entropy Transport Problems
In this paper, we introduce weak optimal entropy transport problems that cover both optimal entropy transport problems and weak optimal transport problems introduced by Liero, Mielke, and Savaré
Mather Measures and Ergodic Properties of Kantorovich Operators associated to General Mass Transfers
We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an


Weak transport inequalities and applications to exponential and oracle inequalities
We extend the dimension free Talagrand inequalities for convex distance using an extension of Marton’s weak transport to other metrics than the Hamming distance. We study the dual form of these weak
Optimal transportation under controlled stochastic dynamics
We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and
A characterization of dimension free concentration in terms of transportation inequalities
The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost
A saddle-point approach to the Monge-Kantorovich optimal transport problem
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to $c$-conjugates. A new abstract characterization of the optimal plans is obtained in
Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Abstract We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6 , 587–600) for the Gaussian measure, are implied by logarithmic Sobolev
Displacement convexity of entropy and related inequalities on graphs
We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path
Ricci curvature for metric-measure spaces via optimal transport
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the
Concentration Inequalities for Convex Functions on Product Spaces
Let µ = µ 1 ⊗ … ⊗ µ n denote a product probability measure on ℝ n with compact support. We present a simple proof to get concentration results for convex functions on ℝ n under µ. We use the infimum
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy
We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott,
A new characterization of Talagrand's transport-entropy inequalities and applications
We show that Talagrand's transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an