A course on stochastic processes with an emphasis on the path propeties of Brownian motion. Covers the construction, modulus of continuity, reflection principle and Skorohod's representation for the Brownian motion, its relation with PDEs via Dirichlet problem and with random walks via Donsker's invariance principle. Provides an introduction to potential theory, as well as other relevant aspects of the general theory of Markov processes. Also to deal with path properties of the Brownian motion that are related to geometric measure theory, such as Hausdorff and Packing dimensions and measures.
Prerequisites: Students should be comfortable with integration and measure theory at the level of Math205A/Stat310A, or have done extremely well in a non-measure theoretic class on Brownian motion, such as Stat218. Command of martingale theory at the level of Stat310b/Stat310c is a plus, but not a must.
Revised Text: An Invitation to Sample Paths of Brownian Motion, by Yuval Peres. (please notify me of typos, a better version is to be available soon).
Supplementary text: Revuz and Yor, Continuous Martingales and Brownian Motion, Third Edition.
Meeting: Stat. 200, M 1:15-2:15 (most likely), 2:45 - 4:00, Th 2:45-4:00
Instructor: Amir Dembo, Stat. 129, office hours M 2:15-3:15, e-mail amir@math.stanford.edu phone 725-2237.
TA: Wenzhi Li, Stat. 108, office hours Tu 4:00-5:00, W 4:00-5:00, e-mail wenzhi@stat.stanford.edu phone 725-6162.
Grading : (40%) Attendance score - requires being in ALL three student presentations and 13 out of 19 regular classes. (40%) Presentation of material -- 3 groups of 3-4 students shall jointly prepare one class. (20%) Notes taken of material not in text (student presentations and other material, as announced in class). Neither exams nor homeworks.
Syllabus (from Peres' notes):
3/26 M(--) Tu(1-3) Th(4-5) 4/3 M(--) Tu(6) Th(TA:8-9) 4/10 M(10-11.1) Tu(11) Th(13; 12 to read) 4/17 M(14) Tu(15) Th(16) 4/24 M(17) Tu(17-20) Th(TA:18-19) 5/1 M(20) Tu(21) Th(TA:22) 5/8 M(23) Tu(24) Th(25) 5/15 M(--) Tu(---) Th(---) 5/22 M(--) Tu(Students) Th(Students) 5/29 M(--) Tu(Students) Th(---)
See also seminar for current activity in related areas.