Orateur
Description
Functional inequalities (Poincaré, log-Sobolev, etc.) represent a ubiquitous tool in probability, since they help us to quantify the convergence to equilibrium of ergodic Markov processes and imply good concentration properties of a probability measure, among other properties. It is natural to wonder if these inequalities remain valid if we perturb the measure. It is known that if there exists a globally Lipschitz map pushing forward the source measure towards its perturbation, then it is easy to transport certain functional inequalities. For example, Caffarelli’s contraction theorem states that the optimal transport map between the Gaussian measure and a log-concave perturbation has the desired Lipschitz regularity.
How could Caffarelli's theorem be extended? Is it possible to do this in more general spaces? Is the optimality of the transport map necessary for this purpose? In this talk I will partially answer to these questions: I will show how a Lipschitz transport map exists if we consider log-Lipschitz perturbations of a measure on a Riemannian manifold, via an alternative to the optimal transport map, induced by the Langevin diffusion associated to the source measure (aka Kim-Milman’s heat flow transport map).
