# Geometry and Probability

### Math 545 - Topics in Riemannian Geometry - Fall 2001

Many applications of probability theory to science and engineering naturally lead to the study of stochastic processes on general manifolds. For example, the diffusion of large molecules in solution or the tracking of an aircraft in flight may both be approached by considering processes on the Lie group of euclidian motions in 3-space. At the same time, Brownian motion has been used as a tool to study purely geometric problems in Riemannian geometry (particularly on negatively curved spaces).

The purpose of the course is to provide an introduction to stochastic processes on Riemannian manifolds, without assuming prior knowledge in differential geometry or advanced probability. I will assume familiarity with multivariable calculus and the basic ideas in measure theory, although some of the background material will be recalled as needed.

Note: It is not necessary to have taken Math 441-2 (Geometry and Topology) previously. If you plan on taking those qualifying courses later or concurrently, the present one could be a helpful "warm up".

Notes for the course can be downloaded from here. Another source is "An Introduction to the Analysis of Paths on a Riemannian Manifold" by Daniel Stroock.

I expect to cover the following topics:

1. Stochastic calculus on the real line;
2. basic Riemannian geometry: geodesics, parallel translation, normal coordinates, Jacobi fields, Lie groups and symmetric spaces, etc;
3. manifold-valued semimartingales and quadratic variation;
4. connections and martingales;
5. integration along a process, Ito and Stratonovich integrals;
6. Brownian motion and diffusions on a Riemannian manifold, short time asymptotics;
7. some stochastic differential equations on manifolds;
8. diffusion limits of random flights and central limit theorems;
9. if time permits, we may discuss some geometric topics about nonlinear filtering, along the lines of R. W. R. Darling's articles Geometrically Intrinsic Nonlinear Recursive Filters, I and II.

Classes will be held in Cupples I 199 on Tuesdays and Thursdays from 2:30 till 4:00. If you have questions, please contact me at feres@math.wustl.edu or in Cupples I room 17 (phone 5-6752). Please note that I'll be out of town until August 6.