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.
"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.
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.
A PDF version of this outline is available here.
- 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.