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Publications Influence

Topics in Optimal Transportation

- C. Villani
- Mathematics
- 1 March 2003

Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric… Expand

3,511 450

Optimal Transport: Old and New

- C. Villani
- Mathematics
- 17 November 2008

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical… Expand

3,681 448

Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

- F. Otto, C. Villani
- Mathematics
- 1 June 2000

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… Expand

718 116- PDF

Ricci curvature for metric-measure spaces via optimal transport

- J. Lott, C. Villani
- Mathematics
- 7 December 2004

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… Expand

898 101- PDF

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

- C. Villani
- Physics
- 31 December 2002

This text has appeared with minor modifications in the Handbook of Mathematical Fluid Dynamics (Vol. 1), edited by S. Friedlander and D. Serre, published by Elsevier Science (2002). I have performed… Expand

683 56- PDF

On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations

- C. Villani
- Physics
- 1 September 1998

This paper deals with the spatially homogeneous Boltzmann equation when grazing collisions are involved.We study in a unified setting the Boltzmann equation without cut-off, the Fokker-Planck-Landau… Expand

289 43- PDF

Entropy Dissipation and Long-Range Interactions

- Radjesvarane Alexandre, L. Desvillettes, C. Villani, B. Wennberg
- Mathematics
- 1 June 2000

Abstract:We study Boltzmann's collision operator for long-range interactions, i.e., without Grad's angular cut-off assumption. We establish a functional inequality showing that the entropy… Expand

217 40

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates

- J. Carrillo, R. McCann, C. Villani
- Mathematics
- 31 December 2003

The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4]… Expand

389 37- PDF

On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation

- L. Desvillettes, C. Villani
- Mathematics
- 1 February 2005

As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the… Expand

296 37- PDF

Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces

- F. Bolley, A. Guillin, C. Villani
- Mathematics
- 7 March 2005

We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle… Expand

185 21- PDF