V4F2 Markov Processes
Tuesday 16.15-17.45 and Thursday 10.15-11.45, Zeichensaal, Wegelerstr. 10.
Tutorial classes: Nikolay Barashkov, Robert Crowell / Group 1: Mon 12-14 N0.007(Neubau) and Group 2: Wednesday 8-10 SemR 1.008.
Exam: to be fixed.
- Basic theory of Markov processes, strong Markov property.
- Markov chains in discrete time (Generator, martingales, recurrence and transience, Harris Theorem, ergodic averages, central limit theorem)
Basic measure theory, conditional expectations, discrete time martingales, Brownian motion.
The lecture notes of Prof. Bovier SS2017 foundations course on Stochastic Processes are available here (pdf). There you find all the necessary background material.
The first part of the course will be mainly based on Prof. Eberle's lecture notes for Markov processes WS16/17 (pdf) and on Ligget's book. Some notes for the lectures will be posted here:
- Note 1 : Introduction, examples, the canonical setup and the strong Markov property. (pdf) [version 1.1, posted 24/10/2017]
- Note 2 : Martingale connection, recurrence of discrete Markov chains, Forster-Lyapounov criteria for recurrence. [version 1, posted 2/11/2017]
- Ligget: Continuous Time Markov Processes, AMS
- Chung: Lectures from Markov Processes to Brownian Motion, Springer
- Eberle: Lecture notes for the course "Markov Processes"
- Sheet 1 (due on thursday 2/11, collected during the lecture)
- Sheet 2 (due on tuesday 7/11, collected during the lecture)
- Sheet 3 (due on tuesday 14/11, collected during the lecture)
- Sheet 4 (due on tuesday 21/11, collected during the lecture)
- Lecture 10/10 : Overview of the course. Definition of a Markov process. Transition kernels.
- Lecture 12/10 : Contruction of a Markov process via Kolmogorov's theorem. General Markov property via the shift operator. Examples.
- Lecture 24/10 : Other examples of Markov processes: Brownian motion and the Poisson process. The need for the canonical setup. Feller property and the right continuous filtration.
- Lecture 26/10 : Stopping times. The Strong Markov property. Examples where the strong Markov property does not hold.
- Lecture 2/11 : Zeros of the Brownian motion. Martingale problems for Markov chains.
- Lecture 7/11 : Lyapounov functions, recurrence properties via supermartingales of the Markov chain.
- Lecture 9/11 : Recurrence and transience of discrete Markov chains. Classification of states and irreducibility. Forster-Lyapounov criteria for recurrence.
- Lecture 14/11 : Recurrence for chains in general state spaces. Weak convergence in Polish spaces. Existence of invariant measures.
- Lecture 16/11 :