**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