Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Problems in Brownian motion (Fall 2011)

Mon 11:00 - 12:30, in N25

About the course: This is an (unofficial) first course in Brownian motion, but there will be no lectures. The course will be entirely devoted to solving problems (by students) and reading up on whatever concepts are needed to solve them. I shall select topics and problems and comment on solutions. Everyone who attends will be expected to attempt most problems and present solutions to many in the weekly meetings. The list of problems from the book of Mörters and Peres can form the backbone but we shall also borrow from Durrett and Karatzas & Shreve and elsewhere. There are no credits for this course. The basic idea is that in a usual course, one spends class-time on the big picture and theorems, and expects that students will practise solving problems by himself/herself. In contrast, here we shall only discuss problems, their solutions and techniques, and leave the reading to the student. The book of Pólya and Szegö is a model for how far one can go this way. A course in measure theoretical probability is a necessary and sufficient prerequisite.

Tentative list of topics:
  • Construction of Brownian motion and its continuity properties (ch. 1 in [MP])
  • Markov and strong Markov prperties and many consequences (ch. 2 in [MP])
  • Stochastic integration (ch. 7 in [MP])
  • Brownian motion and random walks (ch. 5 in [MP])
  • Connections to harmonic functions and PDEs (ch. 3, ch. 8 in [MP])

Weekly progress:
Week 1 06 Feb Construction. Dudley's theorem for Gaussian processes. Problem set
Week 2 13 Feb Continuity properties. Paley-Wiener-Zygmund and Dvoretsky. Problem set
Week 3 22 Feb Absolute continuity and singularity. Cameron Martin. Problem set
Week 4 28 Feb Series representation of BM. Problem set Bernstein's theorem
Week 5 12 Mar Markov and strong markov properties Problem set
Week 6 19 Mar ---
Week 6 26 Mar Strong Markov property, running maximum Problem set
Week 6 02 Apr Martingales in Brownian motion Problem set