Math 553 - Brownian motion on Riemannian manifolds

Spring 2010

Section information: Classes meet on Tuesdays and Thursdays from 1:00PM to 2:30PM, in Cupples I room 207.

Subject: This is an introduction to stochastic calculus on Riemannian manifolds with a focus on the local analysis of Brownian motion and diffusions processes. Familiarity with the basic facts covered in a first semester in manifold theory as well as basic measure theory is assumed. The necessary background in Riemannian geometry will be provided. Majors from other disciplines who may not have taken these math courses but have a strong background in probability theory and feel sufficiently motivated are welcome to join.

Text: An introduction to the analysis of paths on a Riemannian manifold, by Daniel W. Stroock. American Mathematical Society, Mathematical Surveys and Monographs, Volume 74.

Tentative list of topics:

• Brownian motion and diffusions on Euclidian space;
• Riemannian manifolds;
• Brownian motion and diffusions on Riemannian manifolds;
• Elements of stochastic differential equations on manifolds;
• Local analysis of Brownian motion;
• Further topics as time permits (e.g., the Feynman-Kac formula and other PDEs.)

Coursework: Grades will be based on homework assignments and/or a presentation at the end of the course.

Notes: An old set of notes containing a lot of background material in geometry and probability can be downloaded here.