The main goal of the programme is to study a fundamental object in probability theory and stochastic processes—finite state Markov chains—and to tackle some questions whose solutions are not discussed in the literature.
In the first (and main) part of the programme, we plan to investigate the following subjects:
- Conditional measure and expectation
- Definition of a Markov process
- Stationary and ergodic measures
- Stopping times and strong Markov property
- Perron–Frobenius theorem and strongly recurrent Markov processes
- Döblin coupling approach
- Kolmogorov classification of Markov chains
- Law of large numbers and mixing
- Generalities on large deviations
- Large deviation principle for Markov processes
After completing the study of these questions, the students will work in groups on two or three clearly defined projects. This will allow them to deepen their knowledge of the subject and to acquire an experience in carrying out research in mathematics.