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