Prerequisites

Elementary probability (as in e.g. the first chapter of Grimmett and Welsh). Basic abstract analysis and measure theoretic probability (as given e.g. in the mastermath course ``measure theoretic probability’’).

Content of the course
This course provides and introduction in the theory of interacting particle systems, a class of interacting Markov processes used to model many real-world phenomena such as the spread of an infection (contact process), the evolution of opinions in a population (voter model), transport phenomena, and (non)-equilibrium statistical physics.
The course treats the following basic techniques

  • Markov processes in continuous time, semigroups, generators, invariant and ergodic measures, martingales. Elements of the construction of a general class of interacting particle systems.
  • Coupling, monotonicity and positive correlations
  • Duality
  • These basis techniques will then be applied in the context of the exclusion process, a basic interacting particle system used e.g. to model transport of molecules,  and traffic.
  • Depending on time we will treat the hydrodynamic limit (derivation of the heat equation starting from the exclusion process).

Aims
1. Learn the basic techniques of interacting particle systems and Markov process theory.
2. Being able to apply these techniques in the exclusion process and related models.
3. Being able to read a research paper with up to date techniques in this area and give a presentation about it.

Educational method
Two hours of lecture per week, and two hours of time to start reading the project paper (in the beginning we might start with two week of 4 hours lecture, so that the basic techniques are sufficient to start reading)