Bruno Nazaret

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in

From Poincaré to Logarithmic Sobolev Inequalities: A Gradient Flow Approach

{\mathbb{R}^d}

Best constant in Sobolev trace inequalities on the half-space

. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous

On the Bakry-Emery criterion for linear diffusions and weighted porous media equations

{W^{-1,p}_\gamma}

Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

-Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

Among R 3 -valued triples of random vectors (X,Y,Z) having fixed marginal probability laws, what is the best way to jointly draw (X,Y,Z) in such a way that the simplex generated by (X,Y,Z) has maximal average volume? Motivated by this simple question, we study optimal trans- portation problems with several marginals when the objective function is the determinant or its absolute value. Mathematics Subject Classification. 28-99, 49-99.