Math 784 - Brownian Motion

MATH 784 - Topics in Probability (Brownian Motion)

Winter 2013 -- Course syllabus

Professor

Louigi Addario-Berry | louigi@math.mcgill.ca | Tel: (514) 398-3831 (office) | 1219 Burnside Hall | Office Hours: Tuesday and Thursday, 9:30-11:00 or by appointment

Time and Location

To be determined in consultation with students. We will have a planning session on Tuesday, Jan 8, to schedule the class times for the remainder of term. Please register for the course or else email me if you wish to be informed about the time/location of the planning session.

Course book

Morters and Peres, Brownian Motion

Course Outline

This course will cover a selection of advanced topics in Brownian motion, including (time permitting):

Construction of Brownian motion, basic properties of Brownian sample paths.
Brownian motion as a Markov process; Brownian motion as a martingale.
The law of the iterated logarithm
Donsker's invariance principle, arcsine laws
Recurrence and transience, occupation measures and Green's functions
Hausdorff dimension of (subsets of) Brownian motion sample paths
Brownian local time
Stochastic integrals with respect to Brownian motion; Tanaka's formula; Feynman-Kac formula

We may also cover some of: potential theory of Brownian motion, intersections of Brownian paths, exceptional sets for Brownian motion, and an introduction to SLE. The course will be based on the excellent recent book on Brownian motion, by Peter Mörters and Yuval Peres. Prerequisites: Math 587 and parts of Math 589 or equivalent. In particular, comfort with convergence in distribution and with martingale convergence theorems are important. Some experience with Markov processes/Markov chains would also be an asset. However, I encourage all interested students to contact me to discuss their background and whether the course is appropriate for them.

Grading Scheme

The grade will be based on an in-class presentation and written literature review on a topic related to Brownian motion. Topics will be selected in individual consultation with each student.

Additional Information

In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/students/srr/honest/ ) for more information).

	MATH 784 - Topics in Probability (Brownian Motion) Winter 2013 -- Course syllabus

Professor	Louigi Addario-Berry \| louigi@math.mcgill.ca \| Tel: (514) 398-3831 (office) \| 1219 Burnside Hall \| Office Hours: Tuesday and Thursday, 9:30-11:00 or by appointment
Time and Location	To be determined in consultation with students. We will have a planning session on Tuesday, Jan 8, to schedule the class times for the remainder of term. Please register for the course or else email me if you wish to be informed about the time/location of the planning session.
Course book	Morters and Peres, Brownian Motion
Course Outline	This course will cover a selection of advanced topics in Brownian motion, including (time permitting): Construction of Brownian motion, basic properties of Brownian sample paths. Brownian motion as a Markov process; Brownian motion as a martingale. The law of the iterated logarithm Donsker's invariance principle, arcsine laws Recurrence and transience, occupation measures and Green's functions Hausdorff dimension of (subsets of) Brownian motion sample paths Brownian local time Stochastic integrals with respect to Brownian motion; Tanaka's formula; Feynman-Kac formula We may also cover some of: potential theory of Brownian motion, intersections of Brownian paths, exceptional sets for Brownian motion, and an introduction to SLE. The course will be based on the excellent recent book on Brownian motion, by Peter Mörters and Yuval Peres. Prerequisites: Math 587 and parts of Math 589 or equivalent. In particular, comfort with convergence in distribution and with martingale convergence theorems are important. Some experience with Markov processes/Markov chains would also be an asset. However, I encourage all interested students to contact me to discuss their background and whether the course is appropriate for them.
Grading Scheme	The grade will be based on an in-class presentation and written literature review on a topic related to Brownian motion. Topics will be selected in individual consultation with each student.

Additional Information	In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded. McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/students/srr/honest/ ) for more information).