The subject of this course 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 course, I will discuss several of the most important classes of models, including voter models, contact processes, and exclusion processes. I will begin by discussing the construction problem (including a treatment of semigroups and generators) and some of the main tools used in the field (including coupling and duality).
Much of the material will come from my two books:
(1) "Interacting Particle Systems", Springer "Grundlehren" series #276 (1985) (reprinted in 2004 in their "Classics in Mathematics" series),
(2) "Stochastic Interacting Systems", Springer "Grundlehren" series #324 (1999).
(I have asked that both books be placed on reserve in both the Mathematics Reading Room, and the EMS Library. I also have several extra copies of the 1985 book, which I am willing to lend out).
(3) Four years ago, I wrote some introductory lecture notes for a school in Trieste, Italy. I will distribute copies of these notes in class.
In order to follow the lectures, one 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.
I expect that there will be both graduate students and some faculty in the course. I will try to teach it in such a way that both groups can benefit from it. Before each lecture, I will list below the topic of the lecture, and a more detailed reference for it. Students are expected to attend all lectures. Faculty are welcome to come or not according to the topic to be covered.
There will be occasional problems assigned. HW 1 is due January 20. HW 2 is due February 6. HW 3 is due February 27. HW 4 is due March 15. HW 5 is due April 24. HW 6 is due May 22.