**Markov Processes, Winter term 2016/17**

Tuesdays 12.15-14.00 and thursdays 12.15-14.00, Kleiner Hörsaal, Wegelerstr. 10

**Lecture course:** Andreas Eberle

**Exercises:** Raphael Zimmer

**Tutorial classes:** Claudio Bellani, Daniel Koenen

**Exam:** oral. First attempt: 15.2., 22.2., 2.3. Second attempt: 29.3.

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

**Prerequisites:** Conditional expectations, martingales, Brownian motion.

My lecture notes of the 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. Please let me know any corrections (small or large) !

**Further Material**:

- 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 24.10.)
- Sheet 2 (hand in until 31.10.)
- Sheet 3 (hand in until 7.11., Correction in 4b) (2): v(x) statt u(v))
- Sheet 4 (hand in until 14.11., Correction in 3b): V(x)=log(|x|)^a
- Sheet 5 (hand in until 21.11.)
- Sheet 6 (hand in until 28.11.)
- Sheet 7 (hand in until 5.12., Correction in 2d): the eigenvalues correspond to -L)
- Sheet 8 (hand in until 12.12., Correction in 3b): Rotational symmetry is required)
- Sheet 9 (hand in until 19.12.)
- Sheet 10 (hand in until 9.1., corrected definition of coupling time)
- Sheet 11 (hand in until 16.1.)
- Sheet 12 (hand in until 23.1.)
- Sheet 13
- Sheet 14 (added correction in Exercise 1)

February 2017 Andreas Eberle eberle@uni-bonn.de