V5B2 - Selected Topics in Analysis and PDE - Dispersive PDEs: deterministic and probabilistic perspectives
Summer Semester 2021
- Dr. Leonardo Tolomeo
- Tue 14-16 on zoom
Every Tuesday there will be a lecture, taking place on Zoom. The handwritten notes taken during the lectures will be made available on this website. Recordings of the lectures are available, please contact Dr. Tolomeo via email if you are interested. For convenience, the recording of lecture 1 (and only lecture 1) is available on this page.
- Notes of lecture 1, Recording of Lecture 1,
- Notes of lecture 2,
- Notes of lecture 3,
- Notes of lecture 4,
- Notes of lecture 5,
- Notes of lecture 6,
- Notes of lecture 7,
- Notes of lecture 8,
- Notes of lecture 9,
- Notes of lecture 10,
- Notes of lecture 11,
- Notes of lecture 12,
- Notes of lecture 13.
The information about the zoom session can be found on Basis.
TopicsThis course aims at providing the basis for the study of dispersive equations, both in the deterministic setting, and in the probabilistic one. Our goal is to show how probabilistic effects affect the behaviour of these equations, greatly improving the results available. We will cover
1. Strichartz estimates for Schrödinger and wave equations.
2. Local well posedness theory Schrödinger and wave equations in subcritical Sobolev spaces H^s.
3. Global well posedness theory for Schrödinger and wave equations in H^1.
4. Ill posedness in supercritical Sobolev spaces.
5. Local well posedness in supercritical Sobolev spaces for random initial data.
6. Global well posedness for random initial data.
If time allows, we will also discuss some features of the associated stochastic PDEs.
PrerequisitesAnalysis: basic real and complex analysis, basic knowledge of Fourier analysis.
Probability: measure theoretical approach to probability, Gaussian random variables, independence.
- L. Grafakos, Classical and modern Fourier analysis.
- T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis.
- M. Gubinelli, T. Souganidis, N. Tzvetkov, Singular Random Dynamics, Chapter 4.