Math 833 - Continuous Time Markov Processes
Meetings: TR 11:00-12:15, Van Vleck B131
Instructor: Benedek Valkó
Office: 409 Van Vleck
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
Liggett: Continuous Time Markov Processes - An Introduction
A couple of other useful references:
- Ethier, Kurtz: Markov Processes - Characterization and
- Norris: Markov Chains
- Komorowski, Landim, Olla: Fluctuations in Markov
Processes - Time Symmetry and Martingale Approximation
We might also use additional resources.
We will cover the Chapters 2-4 from Liggett:
- 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
The final grade will be based on homework assignments. There is no
final exam in the course.
First assignment: Problems 2.32, 2.36, 2.55, 2.65 from
Liggett. Due date: February 27
First assignment: hw2 Due date:
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