Pour tout renseignement complémentaire, veuillez contacter les
organisateurs, Hakim
Boumaza, Mathieu
Lewin
ou Stéphane
Nonnenmacher.
| 14h - 15h |
Antoine Mouzard (Paris Nanterre) |
Schrödinger operator with distributional potential and Strichartz inequalities Abstract: In this talk, I will present an approach to study Schrödinger operators with distributional potentials. The motivation comes from stochastics PDEs with the Anderson operator which is the Schrödinger operator with white noise potential. I will explain how paracontrolled calculus allows the construction and study of the operator and associated evolution PDEs with the example of the nonlinear Schrödinger equation. On the circle, we are able to consider generic rough potentials of Hölder regularity greater than one and prove invariance of the associated Gibbs measure, following works from Lebowitz, Rose and Speer and Bourgain. On compact surface, even the construction of the Anderson Hamiltonian requires a probabilistic renormalization procedure. Based on joint works with A. Debussche and I. Zachhuber. |
| 15h15 - 16h15 | Benoit Dagallier (Paris Dauphine) |
Stochastic dynamics and the Polchinski equation
Abstract: I will discuss a general framework to obtain large-scale properties of statistical mechanics and field theory models. A well known idea is to introduce a dynamics that samples from the model and controls its long time behaviour. The Langevin dynamics is a popular and successful choice, but is hard to use in a number of cases as I will explain. In this talk I will introduce another object, the Polchinski dynamics, based on renormalisation group ideas. This dynamics has appeared in very different contexts in connection with convex analysis (it is the same as Eldan's stochastic localisation) and optimal transport. I will motivate the construction of the dynamics from a physics perspective and explain how it can be used to prove functional inequalities (Poincaré, log-Sobolev) via a generalisation of Bakry and Emery's convexity argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau, and the review paper https://arxiv.org/abs/2307.07619 |