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
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.
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
Non-classical stochastic flows (coalescence, splitting,
stickiness) and their discrete counterparts. Stability/sensitivity.
- Independent increments.
- Stochastic integration: Wiener chaos.
- Brownian rotations via stochastic
- More on Brownian rotations.
- Stochastic flows.
- Harris flows as Brownian rotations.