In this seminar, we will go through material from the area of Interacting Particle Systems. This subject is a part of probability theory that is used to model situations that arise in many areas, including physics, biology, economics, and even sociology. Typically, in the resulting models, there are an infinite number of agents (atoms, bacteria, people...) that change their states in continuous time, according to stochastic rules that involve dependence among the agents. Without this dependence, the state of each agent would evolve according to a finite or countable state Markov chain. With the dependence, one has to keep track of the states of all the agents, and this results in a Markov process on an uncountable space of configurations. Typical questions are: What are all the stationary distributions for the process? Does the limiting distribution (as time tends to infinity) exist, and if so, how does this limit depend on the initial distribution?

In this seminar, we will cover several of the most important classes of models, starting with exclusion processes.

On January 9, we will get organized, and I will give a brief introduction to the subject. Participant lectures will come from

1. some lecture notes I wrote several years ago,

2. my first book "Interacting Particle Systems", Springer "Grundlehren" series #276 (1985) (reprinted in 2004 in their "Classics in Mathematics" series), and

3. hopefully some recent papers.

Participants should have a good grounding in graduate level real analysis and probability theory. It is particularly important to be comfortable with Markov chains and martingales.