Markov Processes, Winter term 2014/15
Tuesdays 12.15-14.00 and thursdays 12.15-14.00, Kleiner Hörsaal, Wegelerstr. 10
Lecture course: Andreas Eberle
Exercises: Lisa Hartung
Tutorial classes: Jörg Martin
Exam: oral (12.2, 25.2., 27.2.), second round 26.3.
The course will cover a part of the following topics:
- Markov chains in discrete time (Generator, martingales, recurrence and transience, Harris Theorem, ergodic averages, central limit theorem)
- Markov chains in continuous time (Construction, forward and backward equations, martingale problem, ergodicity, interacting particle systems on finite graphs)
- General Markov processes (Semigroups and generators, Feller and L2 approach, martingale problem, Brownian motion with boundary and absorption, h transform, diffusions, interacting particle systems on Zd)
- Long time behaviour (ergodicity, couplings and contractivity, L2 theory, Dirchlet forms, functional inequalities, phase transitions)
- Limits (Weak convergence, Donsker invariance principle, diffusion limits, limits of martingale problems, scaling limits of interacting particle systems, large deviations)
Some of the topics may also be treated in a follow up course in summer term.
Prerequisites: Conditional expectations, martingales, Brownian motion.
The lecture notes of last semester's foundations course on Stochastic Processes are available here. There you find all the necessary background material. Alternatively, you may consult the more compact book Probability Theory by Varadhan. Sections up to 5.5 have been covered in previous courses and will be assumed. Section 5.7 and Chapter 6 will be covered in this course.
Lecture Notes: The most recent version of the lecture notes is available here in uncorrected form. Please let me know any corrections (small or big) !
Further Material:
- Uncorrected lecture notes from 2008
- Liggett: Continuous-time Markov processes
- Stroock: An introduction to Markov processes
- Pardoux: Markov processes and applications
- Ethier/Kurtz: Markov processes: Characterization and convergence
- Bass: Stochastic processes
- Bakry/Gentil/Ledoux: Analysis and geometry of Markov diffusion operators
- Meyn/Tweedie: Markov chains and stochastic stability
- Brémaud: Markov chains
- Levin/Peres/Wilmer: Markov chains and mixing times
- Varadhan: Probability Theory
- Kipnis/Landim: Scaling limits of interacting particle systems
- Hairer: Convergence of Markov processes (Lecture notes)
- Malrieu: Processus de Markov et inégalités fonctionelles (Lecture notes)
- Lindgren: Lectures on stationary stochastic processes (Lecture Notes)
Simulations (Mathematica Notebooks):
Problem Sheets:
- Sheet 1 (hand in until 13.10.)
- Sheet 2 (hand in until 20.10./24.10.)
- Sheet 3 (hand in until 31.10.)
- Sheet 4 (hand in until 7.11.) (Correction in Ex. 1 b): the assumption W~-W is required)
- Sheet 5 (hand in until 14.11.)
- Sheet 6 (hand in until 21.11.)
- Sheet 7 (hand in until 28.11.)
- Sheet 8 (hand in until 5.12.) (Correction in Ex. 2: 1+delta and 1-delta have to be interchanged)
- Sheet 9 (hand in until 12.12.)
- Sheet 10 (hand in until 9.1.)
- Sheet 11 (hand in until 23.1.)
December 2014 Andreas Eberle eberle@uni-bonn.de