The use of the entropic regularization and the relative Sinkhorn algorithm in order to calculate optimal transport problems is nowadays widely used.

We propose a new viewpoint on application of this regularization technique to variational mean-field games with quadratic Hamiltonian. In particular we focus on the fact that this regularization amounts to solving second order variational mean-field games, dropping in particular the requirement for the regularization parameter to be small, whenever we are interested in the case with diffusion.

Based on a joint work with J.-D. Benamou, G. Carlier and L. Nenna.

The Dysthe equation is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. After reviewing the derivation of the Dysthe and related equations, we will focus on the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$.

Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S $ and its dual $V^2_S $ of square bounded variation functions.

We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s\gt 0$.

Our results extend to the finite depth version of the Dysthe equation.

This talk is based on a joint work with Razvan Mosincat (University of Bergen) and Jean-Claude Saut (Université Paris-Saclay).

The Caffarelli contraction theorem states that optimal transport maps (for the quadratic cost) from a Gaussian measure onto measures that satisfy certain convexity properties are globally Lipschitz, with a dimension-free estimate. It has found many applications in probability, such as concentration and functional inequalities. In this talk, I will present an alternative proof, using entropic interpolation and variational arguments. Joint work with Nathael Gozlan and Maxime Prod'homme.

Many stochastic systems can be viewed as gradient flow ('steepest descent') in the space of probability measures, where the driving functional is a relative entropy and the relevant geometry is described by a dynamical optimal transport problem. In this talk we focus on these optimal transport problems and describe recent work on the limit passage from discrete to continuous.

Surprisingly, it turns out that discrete transport metrics may fail to converge to the expected limit, even when the associated gradient flows converge. We will illustrate this phenomenon in examples and present a recent homogenisation result.

This talk is based on joint work with Peter Gladbach, Eva Kopfer, and Lorenzo Portinale.

We consider the time-dependent fractional $p$-Laplacian equation with parameter $p\gt 1$ and fractional exponent $0\lt s\lt 1$. It is the gradient flow corresponding to the Gagliardo fractional energy. Our main result is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on delicate analysis. We will concentrate on the sublinear or “fast” regime, $1\lt p \lt 2$, since it offers a richer theory. Fine bounds in the form of global Harnack inequalities are obtained as well as solutions having strong point singularities (Very Singular Solutions) that exist for a very special parameter interval. They are related to fractional elliptic problems of nonlinear eigenvalue form. Extinction phenomena are discussed.