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:

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