Math 598 - Brownian Motion

MATH 598 - Topics in Probability (Brownian Motion)

Fall 2016 -- Course outline

Professor

Louigi Addario-Berry | louigi.addario@mcgill.ca | Tel: (514) 398-3831 (office) | 1219 Burnside Hall | Office Hours: By appointment, or just drop in.

Time and Location

Monday and Wednesday, 10:00-11:30, Burnside 1205.

Course book

Morters and Peres, Brownian Motion. Additional handouts may be provided during the course.

For material on conditional expectation and martingales, see James Norris's Advanced Probability notes.

For my own course notes, see here.

A proof of the covering theorem used in studying Brownian motion occupation measures can be found here.

History

"Extremely minute particles of solid matter, whether from organic or inorganic substances, when suspended in pure water, or in some other aqueous fluids, exhibit motions for which I am unable to account, and which from their irregularity and seeming independence resemble in a remarkable degree the less rapid motions of some of the simplest animalcules of infusions."
        --Robert Brown, 1829.

Course Outline

The theory of Brownian motion is one of the great interdisciplinary success stories of mathematics. After the initial observations by Brown (a biologist) and important, independent contributions by Thiele (statistics), Bachelier (mathematical finance), Einstein and Smoluchowski (physicists) in the period 1880-1910, a rigorous construction was given by Norbert Wiener (mathematician) in 1923. Today, the theory of Brownian motion plays an important role in all these fields, and in many more.
This course will rigorously introduce and describe the fundamental properties of Brownian motion and related stochastic processes, in particular:

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

Some of the following topics will also be addressed, time permitting.

Hausdorff dimensions of (subsets of) Brownian motion sample paths
Polar sets, intersections and self-intersections of Brownian motion:
Fast times and slow times.
The Brownian continuum random tree
Introduction to SLE
Introduction to the theory of continuous martingales.
Introduction to Lévy processes
Itô's excursion theory for Brownian motion.
Gaussian processes, the Gaussian free field.
A PDF version of this outline is available here.

Prerequisites

Math 587 or permission of the instructor.

Grading Scheme

Class participation, 25%; assignments, 75%.

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).

In the event of extraordinary circumstances beyond the University’s control, the content and/or evaluation scheme in this course is subject to change.

	MATH 598 - Topics in Probability (Brownian Motion) Fall 2016 -- Course outline

Professor	Louigi Addario-Berry \| louigi.addario@mcgill.ca \| Tel: (514) 398-3831 (office) \| 1219 Burnside Hall \| Office Hours: By appointment, or just drop in.
Time and Location	Monday and Wednesday, 10:00-11:30, Burnside 1205.
Course book	Morters and Peres, Brownian Motion. Additional handouts may be provided during the course. For material on conditional expectation and martingales, see James Norris's Advanced Probability notes. For my own course notes, see here. A proof of the covering theorem used in studying Brownian motion occupation measures can be found here.
History	"Extremely minute particles of solid matter, whether from organic or inorganic substances, when suspended in pure water, or in some other aqueous fluids, exhibit motions for which I am unable to account, and which from their irregularity and seeming independence resemble in a remarkable degree the less rapid motions of some of the simplest animalcules of infusions." --Robert Brown, 1829.
Course Outline	The theory of Brownian motion is one of the great interdisciplinary success stories of mathematics. After the initial observations by Brown (a biologist) and important, independent contributions by Thiele (statistics), Bachelier (mathematical finance), Einstein and Smoluchowski (physicists) in the period 1880-1910, a rigorous construction was given by Norbert Wiener (mathematician) in 1923. Today, the theory of Brownian motion plays an important role in all these fields, and in many more. This course will rigorously introduce and describe the fundamental properties of Brownian motion and related stochastic processes, in particular: Construction of Brownian motion, basic properties of Brownian sample paths. Brownian motion as a Markov process; Brownian motion as a martingale. Continuity properties, dimensional doubling Donsker's invariance principle, arcsine laws The law of the iterated logarithm Recurrence and transience, occupation measures and Green's functions Brownian local time Stochastic integrals with respect to Brownian motion; Tanaka's formula; Feynman-Kac formulae Some of the following topics will also be addressed, time permitting. Hausdorff dimensions of (subsets of) Brownian motion sample paths Polar sets, intersections and self-intersections of Brownian motion: Fast times and slow times. The Brownian continuum random tree Introduction to SLE Introduction to the theory of continuous martingales. Introduction to Lévy processes Itô's excursion theory for Brownian motion. Gaussian processes, the Gaussian free field. A PDF version of this outline is available here.
Prerequisites	Math 587 or permission of the instructor.
Grading Scheme	Class participation, 25%; assignments, 75%.

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). In the event of extraordinary circumstances beyond the University’s control, the content and/or evaluation scheme in this course is subject to change.