| Lecturer: | G. Giacomin |
| Périod: |
Term 1 |
| ECTS: |
6 |
| Hourly Volume: | 3 hours per week |
This course is an introduction to the general theory of time homogeneous Markov chains. It will start with a detailed review of the theory in discrete time and discrete state space. Then continuous time and/or more general state spaces will then be considered. Some specific examples (random walks, birth and death chains, renewal processes, branching processes) will be developed in detail.
Outline:
1. Discrete-time chains on countable state spaces: construction, transition matrix, communication classes, periodicity.
Recurrence and transience, invariant measures, ergodic theorems.
2. Potential theory: Martingales, potentials associated with a chain.
Reversible case : electrical networks analogy.
3. Discrete-time chains on general state spaces: Harris chains. Recurrence, transience, invariant measures, asymptotic behavior.
4. Continuous-time chains on countable state spaces: Construction, generator, asymptotic behavior.
5. Convergence to diffusion processes: diffusions as Markov processes, Donsker’s Theorem (convergence to Brownian mot!ion), discussion of the general case.
Bibliography:
- F. COMETS, T. MEYRE, Calcul Stochastique et Modèles de Diffusions, Dunod, 3ème edition 2020
- J. F. LE GALL, Intégration Probabilités et Processus Aléatoires, https://www.imo.universite-paris-saclay.fr/~jflegall/IPPA2.pdf
- R. DURRETT, Probability : Theory and Examples, Duxbury Press (1996)
- J. R. NORRIS, Markov Chains, Cambridge 1997
