**Spring 2014**

**Meetings:** TR 11:00-12:15, Van Vleck B131

**Instructor:** Benedek Valkó

**Office:** 409 Van Vleck

**Phone:** 263-2782

**Email:** valko at math dot wisc dot edu

**Office hours:** by appointment

I will use the class email list to send out
corrections, announcements, please check your wisc.edu email
from time to time.

The course will focus on an important class of stochastic
processes: continuous time Markov processes. We will discuss
specific examples (e.g. Brownian motion, Markov processes with
countable state space) and the general theory (semigroup theory,
infinitesimal generators, Hille-Yosida theory). The second part of
the course will provide an introduction to the theory of
interacting particle systems.

733 (old 831) or an equivalent probability course (with measure theory). It would be useful to have some background from discrete Markov chains (632 or 734), but we can review this within the course.

A couple of other useful references:

- Ethier, Kurtz: Markov Processes - Characterization and Convergence
- Norris: Markov Chains
- Komorowski, Landim, Olla: Fluctuations in Markov Processes - Time Symmetry and Martingale Approximation

We might also use additional resources.

- Ch 2: Continuous time Markov chains

- Ch 3: Feller processes
- Ch 4: Interacting particle systems

In the second half of the semester we will look at various other
topics related to interacting particle systems: hydrodinamic
scaling limits, central limit theorems,...

If you have a another topic you would like to learn about: let me
know!

Third assignment:

1. week (1/21, 1/23): Basic definitions, examples, Markov processes on discrete state space

2. week (1/30): Differentiability of the transition function

3. week (2/4, 2/6): Blackwell's example, the minimal solution of the backward equation with a given Q-matrix

4. week (2/11, 2/13): Constructing the distribution of the MC from the Q-matrix, Feller processes, the resolvent

5. week (2/18, 2/20): The generator, the probability semigroup and generator of a Feller process. From Feller process to infinitesimal description.

6. week (2/25, 2/27): From infinitesimal description to Feller process. Closure of an operator.

7. week (3/6): guest lecture by D. Anderson

8. week (3/11, 3/13): martingales and Feller processes, stationary distributions for Feller processes, duality

9. week (3/25, 3/27): superposition of processes, the Feynman-Kac formula, interacting particle systems introduction

10. week (4/1, 4/3): existence theorem for spin systems, ergodicity of spin systems, coupling

11. week (4/8, 4/10): duality in the voter model, stationary measures for the voter model

12. week (4/15, 4/17): a closer look at the contact process, the exclusion process

13. week (4/22, 4/24): stationary distributions of the exclusion process, symmetric and translation invariant cases

14. week (4/29, 5/1): a first look at hydrodynamic limits: the case of independent random walkers