Brownian motions and stochastic flows

2003/2004, sem. 1

Prof. Boris Tsirelson (School of Mathematical Sciences).
Time and place
Monday 16-19 Dan David 204.
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 L2 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.

Syllabus (tentative)

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.

Lecture notes

  1. Independent increments.
  2. Stochastic integration: Wiener chaos.
  3. Brownian rotations via stochastic integrals.
  4. More on Brownian rotations.
  5. Stochastic flows.
  6. Harris flows as Brownian rotations.