# Ma 144B. Probability

### Schedule

MWF 1 pm, Room 122 in Math Building (also known as Building 15)

### About this course

In this course, we explore some of the basic properties of Brownian motion.
Some topics I hope to cover:

- Brownian motion is locally Holder continuous
- Cameron-Martin theorem
- Applications to harmonic functions: recurrence, transience, occupation times
- Conformal invariance and applications to complex analysis
- The dimension of the zero set of Brownian motion is 1/2
- Kaufman's dimension doubling theorem
- Some stochastic calculus and applications to PDEs
- Feynman-Kac formula (and applications)
- Potential theory: characterization of polar sets and capacities

Most of the topics will be selected from the book Y. Peres and P. Mörters:

http://yuvalperes.com/brbook.pdf.

For some additional topics, I will also use Durrett's book "Brownian motion and martingales."