Markov semigroups provide very general models and tools in the analysis of time evolution phenomena and dynamical systems. This course is a modern overview on probabilistic and geometric aspects of Markov semigroups and associated functional inequalities such as log-Sobolev inequalities.
This course focuses on understanding a few topics centered around Markov semigroups and associated functional inequalities. The important topics to master include Markov semigroups, examples of Markov semigroups coming from stochastic differential equations, Poincare inequality, log-Sobolev inequality and isoperimetric inequality.
The main content follows the book "Analysis and Geometry of Markov Diffusion Operators", Bakry, Gentil and Ledoux, 2013.
The official course catalogue page can be found here.
This course is designed to be accessible for first-year master student in mathematics. A solid background on undergraduate probability and analysis is required. Background on stochastic processes and stochastic differential equations is highly recommended.
| Date | Content | Notes |
|---|---|---|
| Mon 17.02. | Introduction to Markov semigroups | |
| Mon 24.02. | Model Examples: Euclidean heat semigroup, Ornstein-Uhlenbeck semigroup | Week 01-02 |
| Mon 03.03. | from OU semigroup to Gaussian Poincaré inequality, log-Sobolev inequality | |
| Mon 10.03. | Implications of PI and LSI, alternative proofs | Week 03-04 |
| Mon 17.03. | Hypercontractivity and spectral gap as reformulation | Week 5 |
| Mon 24.03. | Transport inequalities | Week 6 |
| Mon 31.03. | Isoperimetric inequalities | Week 7 |
| Mon 07.04. | Bakry-Émery criterion and Gamma-calculus | Week 8 |
| Mon 14.04. | (continued) | Week 9 |
| Mon 21.04. | No Lecture | |
| Mon 28.04. | No Lecture | |
| Mon 05.05. | Kannan - Lovász - Simonovits conjecture | Week 10 |
| Mon 12.05. | Optimal transport viewpoint | Week 11 |
| Mon 19.05. | (continued) | Week 12 |
| Mon 26.05. | Discrete space and open problems | Week 13 |