## MATS421 PDE reading seminar, 3 cr, spring 2014

### Announcements

28.2. No seminar!
### Description

The reading seminar introduces stochastic tools for studying potential theory for partial differential equations. We mainly follow the book
Peter Mörters, Yuval Peres

*Brownian motion*.

Cambridge University Press, 2010.

### Prerequisites

The seminar is intended to be accessible also for graduate students outside stochastics (for example from analysis), so in the first meetings we will review the basic theory of Brownian motion.
However, one should recall, for example from Evans' lecture note below, the concepts of probability space and measure, random variable, independence, expectation and distribution.
### Scedule

Seminar on Fridays at 10.15-12.00, MaD355. The seminar kick-off on Fr 24.1.
- 24.1. Mikko Parviainen: Existence and basic properties of Brownian motion (based on Chapter 1)
- 31.1. Eero Ruosteenoja: Sample path and Markov properties of Brownian motion (Chapter 1 and beginning of Ch 2)
- 7.2. Hans Hartikainen: Markov properties of Brownian motion (Chapter 2)
- 14.2. Antti Luoto: Brownian motion and harmonic functions (Chapter 3 beginning)
- 21.2. Tommi Brander: Transience, recurrence and occupation measures (end of Chapter 3)
- 28.2. No seminar!
- 7.3. Mikko Kuronen: Brownian motion and random walk (Chapter 5)
- 14.3. Anni Toivola: Brownian local time (Chapter 6, see also Karatzas, Shreve: Brownian Motion and Stochastic Calculus, Section 3.6)
- 21.3 Tuomo Ojala: Stochastic integration
- 28.3 Benny Avelin: Limits of superharmonic functions along Brownian paths I (Port, Stone: Ch 5)
- 28.3 Benny Avelin: Limits of superharmonic functions along Brownian paths II (Port, Stone: Ch 5), MaD355, 14.15-16
- 4.4. Heikki Seppälä: On the Hausdorff dimension of Brownian motion. (Chapter 4)
- 11.4. Joonas Heino: Stochastic Wiener criterion and boundary regularity of harmonic functions (final talk)

### Instructor

Mikko Parviainen,
mikko.j.parviainen@jyu.fi, MaD306
### Evaluation

Passing the seminar is based on active participation and presentations.
### Korppi

Course data at Korppi
### Literature

- Mörters, Peres: Brownian motion

- Evans: An introduction to stochastic differential equations
- Port, Stone: Brownian motion and classical potential theory
- Karatzas, Shreve: Brownian Motion and Stochastic Calculus