## TAU:0365.4050 |
## Brownian motions and stochastic flows | ## 2003/2004, sem. 1 |

- Lecturer
- Prof. Boris Tsirelson (School of Mathematical Sciences).
- Time and place
- Monday 16-19 Dan David 204.
- Prerequisites
- Be acquainted with such things as:

probability measures on (the Borel sigma-field of) a compact metric space;

the product of two probability spaces;

the Hilbert space*L*_{2}of square integrable functions on a measure/probability space;

one-parameter semigroups of unitary operators (on a Hilbert space) and their generators;

the central limit theorem of probability theory; convergence of distributions; normal distributions.

- Grading policy
- Written homework and oral exam.

Normal and Poisson distributions as examples of convolution
semigroups on **R**. Corresponding random processes: the
Brownian motion and the Poisson process. Corresponding Hilbert spaces
consist of multiple stochastic integrals.

More general groups: the circle (torus), the group of rotations. Brownian motions in such a group are related to Brownian motions in its tangent space: linearization.

Infinite-dimensional groups (diffeomorphisms, unitary operators etc). Stochastic flows as infinite-dimensional Brownian motions. Linearization.

Non-classical stochastic flows (coalescence, splitting, stickiness) and their discrete counterparts. Stability/sensitivity. Noises.