Markov Processes (M2 - 2026)
Course description
Markov processes are random functions of time whose past and future are conditionally independent, given their present. They are widely used to model complex random evolutions in our physical world, and serve as efficient stochastic algorithms for the exploration of massive networks or the generation of random high-dimensional data. The aim of this course is to provide an introduction to their general theory, with an emphasis on examples and applications. We will cover the following notions:- Discrete-time Markov chains
- Continuous-time Markov chains
- Markov processes on Polish spaces
Bibliography
- Introduction to Markov Processes (Lecture notes in construction, comments welcome!)
- Continuous-time Markov Processes (T. Liggett)
- Markov Chains (J. Norris)
- Markov Processes: Characterization and Convergence (N. Ethier and G. Kurtz)
- Analysis and Geometry of Markov Diffusions Operators (D. Bakry, I. Gentil, M. Ledoux)
- Modern Aspects of Markov Chains (J. Salez)
Practical information
- The lectures take place on Thursdays, 12:00-15:15.
- The exercise sessions take place on Fridays, 12:00-13:30.
- The final exam will take place on January 7, 14:00-16:00 in Room A405.
Tentative planning
- 01.10 - Room D202 - Discrete-time Markov chains: definition and examples
- 02.10 - Room A308 - Strong Markov property; recurrence/transience dichotomy; Polya's theorem
- 08.10 - Course canceled (campus closed)
- 09.10 - Course canceled (campus closed)
- 15.10 - Online via teams - Invariant measures and Ergodic theorem (slides)
- 16.10 - Online via teams - Convergence to equilibrium (slides)
- 22.10 - Room A411 - Continuous-time Markov chains: definition, Markov property, semi-group and generator
- 23.10 - Room A308 - Kolmogorov equations, existence and uniqueness
- 05.11 - Room A308 - Strong Markov property in continuous-time
- 06.11 - Room A308 - Recurrence/transience, invariant measures, ergodic theorem
- 12.11 - Room D203 - Convergence to equilibrium, reversibility, examples
- 13.11 - Room A308 - Markov processes on general spaces: definition, existence
- 19.11 - Room A308 - Convolution semi-groups and Levy processes
- 20.11 - Room A308 - Strong Markov property, Hille-Yosida theorem
- 26.11 - Room D202 - Infinitesimal generator and martingales
- 27.11 - Room A308 - Markov diffusions, Langevin dynamics
- 03.12 - Room A308 - Introduction to mixing times
- 04.12 - Room A308 - Introduction to functional inequalities
- 10.12 - Room C515 - Introduction to the cutoff phenomenon
- 11.12 - Room A308 - Introduction to curvature