Topics in the theory of Markov processes

Doctoral course

Goals

The goal of these lectures is to present some aspects of the theory of Markov processes, with particular emphasis to Ito diffusion processes, both linear and nonlinear. In the first part of the course we will present some elements of the theory of Markov diffusion semigroups: infinitesimal generators, ergodic theory for Markov processes, convergence to equilibrium, functional inequalities, Bakry-Emery theory/Gamma calculus. I f time permit we will discuss about nonlinear diffusion processes of McKean type,  the long time behaviour of solutions to the (forward Kolmogorov) McKean-Vlasov equation and we will study the possible non-uniqueness of invariant measures for such processes.

Sample paths of the Ornstein-Uhlenbeck process from Pavliotis 2014, Sec. 2.4

Teacher

Prof. Greg Pavliotis

 

Organization of the course

Course: Wednesday from 15:15 to 18:00 at CO120

Office hours: Thursday from 13:00 to 15:00 at MA C2 637

 

Lecture notes

 

 

 

Exercices

 

 

Week Series
2
3
4

Bibliography

  • Pavliotis, Grigorios A. Stochastic processes and applications. Diffusion processes, the Fokker-Planck and Langevin equa-tions. Texts in Applied Mathematics, 60. Springer, New York, 2014.
  • Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer, Cham, 2014
  • Bogachev, Vladimir I.; Krylov, Nicolai V.; Röckner, Michael; Shaposhnikov, Stanislav V. Fokker-Planck-Kolmogorov equa-tions.Mathematical Surveys and Monographs, 207. American Mathematical Society, Providence, RI, 2015
  • Dawson, Donald A. Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31 (1983), no. 1,29–85.
  • Chayes, L.; Panferov, V. The Mc Kean-Vlasov equation in finite volume. J. Stat. Phys. 138 (2010), no. 1-3, 351–380.
  • Chazelle, Bernard; Jiu, Quansen; Li, Qianxiao; Wang, Chu Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics. J. Differential Equations 263 (2017), no. 1, 365–397.
  • Long-time behaviour and phase transitions for the Mc Kean–Vlasov equation on the torus J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, A. Schlichting arXiv:1806.01719