Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The members of the ISM Probability Group are involved in research in a broad range of areas spanning theoretical and applied, continuous and discrete probability. A particular focus is on the development and analysis of probabilistic models for real-world phenomena from physics, biology, statistics and computer science. Some specific topics of interest are: statistical physics in a random environment, branching systems in biology, distances and random energy landscapes, data structure analysis using random trees, genetics and population biology.

Many members of the group are also members of the CRM Probability Lab.

- Louigi Addario-Berry (McGill)
- Claude Bélisle (Laval)
- Linan Chen (McGill)
- Luc Devroye (McGill)
- Don Dawson (Carleton)
- René Ferland (UQAM)
- Alexander Fribergh (UdeM)
- Geneviève Gauthier (HEC)
- Chantal Labbé (HEC)
- Sabin Lessard (UdeM)
- Lea Popovic (Concordia)
- Bruce Reed (McGill)
- Wei Sun (Concordia)
- François Watier (UQAM)
- Xiaowen Zhou (Concordia)

Students interested in graduate study in any of the areas cited above are invited to apply for admission to the program. There are no formal prerequisites other than those required by the departments. The following guidelines should be followed, however, and courses selected in consultation with an advisor from the group.

Students in the program are expected to have mastered the subject matter of the undergraduate curriculum in probability theory. All students are required to take the basic courses: Real Analysis Measure Theory and Probability Theory. Students are then expected to take a number of more specialized courses.

Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.

**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.

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.

**Outline: **

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.

Textbook: *Brownian motion*, by Peter Mörters and Yuval Peres, plus other sources as needed.

Prerequisite: Math 587 or permission of instructor.

Espace de probabilité, variables aléatoires, indépendance, espérance mathématique, modes de convergence, lois des grands nombres, théorème central limite, espérance conditionnelle et martingales.

The theory of large deviations studies probabilities of events which, in a large sample, are exponentially rare in the number of samples. We will cover standard methods including large deviations for i.i.d. sequences, occupation measures, and diffusions with small noise. This theory has found many applications in finance and risk management, simulation and sampling, as well as operations research and statistical mechanics. The main part of the course will cover: Introduction to large deviations, the large deviations principle, Sanov's theorem and method of types, Cramer's theorem, Gartner-Ellis theorem, concentration inequalities, large deviations for Markov chains, contraction principle, Varadhan's and Bryc's lemmas, sample path large deviations.

The prerequisites for the course are an advanced course in probability theory and in stochastic processes; including LLNs, CLTs, Markov chains, martingales, Brownian motion.