Seminar on Mathematical Stochastics - Markov processes and their applications

WS 2023/24


Seminar talks: Thursday 10:15 - 11:45, Sed 19, 217

Office hours: Tuesday 17:00 - 18:00, T18


Course in the Moodle system: link (will be available later)


Talks can be given in English or German



SHORT DESCRIPTION OF SEMINAR


Stochastic and Markovian modeling are of importance to many areas of science including physics, biology, engineering, machine learning, as well as economics, finance, and social sciences. One of the basic mathematical tools for such modeling are discrete- and continuous-time Markov processes. They describe random systems evolving in time, whose future behavior depends only on the current stage and is independent of the past.

We will focus on a deeper look into the basic theory of Markov chains and will discuss their applications in physics, genetics, machine learning, etc. In particular, we will cover the following topics:


  • basic concepts of (discrete-time) Markov chains like the Markov property, transition probabilities, hitting probabilities, first return time, classification of states, etc;
  • examples of Markov chains;
  • long-time behavior and stationary distributions;
  • application of Markov processes in population genetics, population theory and machine learning;
  • fluctuations of Markov chains;
  • continuous-time Markov chains and some examples;
  • application in particle statistical mechanics: simple symmetric exclusion process and zero-range process.

    • If you have an idea about the application of Markov chains in your field of interest and are willing to give a talk about that, please contact me by email as soon as possible (preferably until October 9). We will discuss the possibility of including your topic in the schedule.

    The list of topics and rules will be available approx. on October 11 in STiNE.



    REQUIRED KNOWLEDGE


    The seminar requires knowledge of standard courses on probability theory and analysis.



    USED LITERATURE


  • Privault, N. Understanding Markov chains Springer Singapore, Singapore, 2013, x+354
  • Komorowski, T.; Landim, C. & Olla, S. Fluctuations in Markov processes Springer, Heidelberg, 2012, 345, xviii+491
  • Kipnis, C. & Landim, C. Scaling limits of interacting particle systems Springer-Verlag, Berlin, 1999, 320, xvi+442
  • Bach, F. Learning Theory from First Principles April 19, 2023