**Markov Processes, Winter term 2019/20**

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

**Lecture course:** Andreas Eberle

**Exercises:** Florian Kreten

**Tutorial classes on wednesdays:**

Adrian Martini (8-10, N0.008), Florian Kreten (12-14, 1.008), Jason Hu (16-18, 1.008).

**Exam:** oral (**3.2.**, 26.2., 27.2., 18.3.)

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)

**Problem Sheets**

- Sheet 0 (to be discussed in the tutorials during the first week)
- Sheet 1 (due on Monday 14.10.) (Correction in 3a(i): sum k=0 to n)
- Sheet 2 (due on Monday 21.10.) (Correction 2b: Show that (B,X) and (B,S) are Markov, but X is not a Markov process.)
- Sheet 3 (due on Monday 28.10.)
- Sheet 4 (due on Monday 4.11.) (Correction in 2b: "transition probabilities" instead of "transition rates")
- Sheet 5 (due on Monday 11.11.)
- Sheet 6 (due on Monday 18.11.)
- Sheet 7 (due on Monday 25.11.) (New version with small corrections in 1a(vii) and 2e)
- Sheet 8 (due on Monday 2.12.)
- Sheet 9 (due on Monday 9.12.)
- Sheet 10 (due on Monday 16.12.) (added correction in 3a)
- Sheet 11 (due on Tuesday 7.1.) (added correction in 2b and in 4 [deleted one vertex of the tree])
- Sheet 12 (due on Monday 13.1.)
- Sheet 13 (due on Monday 20.1.)

January 2020 Andreas Eberle eberle@uni-bonn.de