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 not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.Date: August 3, 2018. 1991 Mathematics Subject Classification. 60E15, 32F32 and 26D10.

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery [3], the exponential curvature-dimension condition of Bauer et al.[5] and the coarse Ricci curvature of Ollivier [34]. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an L 1 transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD(κ, ∞) of Bakry and Émery [3].

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.

We study the properties of Hopf-Lax formula restricted to convex functions and characterize the optimal transfer plan for weak transport problem.

We give a sufficient and necessary condition for a probability measure µ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto µ. The main tool in the proof is the theory of weak transport costs.As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.Due to its tensorization property the log-Sobolev inequality is a powerful tool and can be used to obtain dimension-free concentration bounds (via the Herbst argument). It has been investigated also in more general settings of Riemannian manifolds and in context of applications to the study of Markov chains, see e.g. the monographs [3,4] and the expository article [11].