FMVE030 Random walks and discrete potential theory

Kursens poäng (högskolepoäng, hp) 7.5
Kursen ges normalt Only once
Tillhör forskarskola Matematik
Tillhör institution Matematiska vetenskaper
Kursansvarig
torbjrn@chalmers.se
Kursbeskrivning
Random Walks is a widely studied field, not at least here in Göteborg. It is both aesthetically appealing and has many useful different applications such as in finance, how infections spread, and material sciences.

Discrete Potential
Theory is an offspring from classical potential theory, which in turn has a long history from problems in physics. In the classical theory, there is a connection to probability via the solution to the Dirichlet problem using Brownian motions. In the discrete setting, the connection is on the other hand evident and there is no real difference between the theories of random walks and discrete potential theory.

Contents
Here is a short list of some of the concepts that will be discussed in the course, but I am of course open to  other paths on request:

* Mean exit time
* Recurrence/transience
* Laplace operator
* Resistance
* Electric networks
* Harnack inequalities
* Green kernel
* Isoperimetric problem
* Transient graphs
* Faber-Krahn inequality
* Feynman-Kac
* Parabolic Harnack inequality
Litteratur
I will, in principle, follow parts of the presentations in:

i) Random Walks and Electric Networks http://arxiv.org/abs/math/0001057, Doyle and Snell, 1984 and 2000
ii) The Art of Random Walks, Andras Telcs, LNM 1885, 2006

But I also plan to add some text from a few of the reference books listed below.

iii) Random walks on infinite graphs and groups http://books.google.com/books/cambridge?ie=UTF-8&vid=ISBN0521552923&refid=ca-print-cambridge&q=&submit=Go, Wolfgang Woess, Cambridge University Press, 2000
Övrigt
http://www.math.chalmers.se/~torbjrn/rw/

Publicerad: ti 29 jan 2013.