Summer School on PDEs and Randomness
Abstracts for the talks
University of Cambridge
Renormalisation and log-Sobolev inequalities
This minicourse introduces an approach to obtain log-Sobolev inequalities for measures appearing in statistical physics and quantum field theory via a renormalisation group approach. I will also mention relations of this approach to the method of stochastic localisation of Eldan and to the variational method of Barashkov--Gubinelli. This minicourse is based on joint works with Thierry Bodineau and Benoit Dagallier.
Introduction to deterministic and random dispersive equations
In this lecture series we discuss linear and nonlinear Schrödinger equations.
Schrödinger equations belong to the class of dispersive equations, which is ubiquitous in mathematical physics and also includes wave equations and the Korteweg-de Vries equation.
In the first half of this lecture series, we present deterministic aspects of Schrödinger equations. In particular, we discuss dispersive estimates, Strichartz estimates, and Bourgain spaces. In the second half, we present probabilistic aspects of Schrödinger equations. In particular, we prove multilinear dispersive estimates for random initial data, which combine both dispersive and probabilistic cancellations. Furthermore, we discuss a recent random tensor estimate of Deng, Nahmod, and Yue, which has already been useful in many applications.
If time permits, we end the lecture series with a brief discussion of important open problems.
Ricci curvature, Bochner inequalities, and stochastic analysis
In this course we will examine the interplay between geometry and stochastic analysis on manifolds. For this we will introduce the Eells-Elworthy-Malliavin calculus in order to characterize two-sided Ricci bounds. Furthermore we will see how this tool applies to Ricci flows and differential Harnack estimates on path space.
Energy solutions and generators of singular SPDEs
Energy solutions provide probabilistic solution theories for singular SPDEs with tractable (quasi-)invariant measures, with the prototypical example being the stochastic Burgers/KPZ equation and white noise invariant measure. Energy solutions were introduced by Gonçalves and Jara and later Gubinelli and they are based on methods from hydrodynamic limits such as replacement lemmas and martingale estimates. More recently, we understood how to use chaos decompositions to construct and control infinitesimal generators in this setting, which leads to a (weak) well-posedness theory of energy solutions. Compared to pathwise approaches like regularity structures, this requires only relatively soft estimates and the method applies to some scaling (super-)critical equations.
In my lectures, I will start with the guiding example of a diffusion in a singular divergence-free vector field, where we can understand the main ideas of energy solutions without many technicalities and we can already see some (super-)critical problems. Then I will present a relatively general and abstract construction of infinitesimal generators, semigroups, and energy solutions based on chaos expansions and infinite-dimensional analysis, due to Gräfner. Finally we will study applications to singular SPDEs.
Date and Location
May 10 - 24, 2023
Max Planck Institute for Mathematics in the Sciences
E1 05 (Leibniz-Saal)
- Rishabh Gvalani
- Francesco Mattesini
- Felix Otto
- Markus Tempelmayr
Administrative ContactKatja Heid
MPI for Mathematics in the Sciences
Contact by Email