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Markov Processes

Module MA4408

This Module is offered by Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2013

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2014WS 2013/4SS 2013

Basic Information

MA4408 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)
Total workloadContact hoursCredits (ECTS)
270 h 90 h 9 CP

Content, Learning Outcome and Preconditions


Markov chains in continuous time, Markov property,
convergence to equilibrium. Feller processes, transition semigroups and their generators, long-time behaviour of the process, ergodic theorems. Applications e.g. to queuing
theory, interacting particle systems or time series.

Learning Outcome

After successful completion of the module, students are able to:
- understand and apply the Markov property
- do calculations with Q-matrices and invariant/reversible distributions
- understand the basics of the theory of Feller processes
- apply ergodic theorems
- analyse the long-term behaviour of a given Markov process.


MA2409 Probability Theory

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 4 Markov Processes [MA4408] de Paula Reis, G. Gantert, N. Mon, 12:15–13:45, BC2 BC2 3.5.06
Wed, 08:30–10:00, BC2 BC2 3.5.06
UE 2 Markov Processes (Exercise Session) [MA4408] de Paula Reis, G. Gantert, N. Mon, 10:15–11:45, BC2 BC2 3.5.06

Learning and Teaching Methods

lecture, exercise module
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.


blackboard, assignments


T. Liggett (2010): Continuous time Markov processes, American Mathematical Society, USA.

Module Exam

Description of exams and course work

The module examination is based on a written exam (90 minutes) or an oral exam (20-30 minutes), depending on the number of attendants. Students have to understand foundations of the Markov property and can adequately apply them in limited time. They are familiar with Feller processes and are able to use ergodic theorems.

Exam Repetition

The exam may be repeated at the end of the semester.

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