Brownian motion

2007/2008, sem. 2

Prof. Boris Tsirelson (School of Mathematical Sciences).
Time and place
Thursday 17-20 Schreiber 209.
Be acquainted with such things as the Hilbert space L2 of square integrable functions on a measure space. Some maturity in analysis is needed. If you did not take "Advanced probability", be ready to accept some theorems without proofs.
Grading policy
Written homework and oral exam.

Lecture notes

  1. Independent increments.
    PDF or Postscript.
  2. Markov and strong Markov.
    PDF or Postscript.
    Corrections: PDF or Postscript.
  3. Random walks and Brownian motion.
    PDF or Postscript.
  4. Brownian martingales.
    PDF or Postscript.
  5. Localization.
    PDF or Postscript.
  6. Time change.
    PDF or Postscript.

Some literature

  1. Durrett R. (Richard), "Probability: theory and examples" (second edition, 1996), Chapter 7.
  2. Durrett R. (Richard), "Brownian motion and martingales in analysis" (1984), Chapters 1-5.

A quote:

"Brownian motion is a process of tremendous practical and theoretical significance. [...] Brownian motion is a Gaussian Markov process with stationary independent increments. It lies in the intersection of three important classes of processes and is a fundamental example in each theory."

Richard DURRETT [1, p. 374].