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Lectures: Monday, Wednesday and Friday, 1pm-2pm, in Science Center 216.

Lecturer: Paul Bourgade, office hours Tuesday and Thursday 10:30am-12pm, you also can email me (bourgade@math.harvard.edu) to set up an appointment or just drop by (Science Center 341).

Lecture notes available here, soon complete. Please let me know about the inaccuracies and typos you will certainly find.

Course description: introduction to continuous stochastic processes, and connections with other mathematical fields (harmonic analysis, concentration of measure, trace formulae).

Prerequisites: you need to be familiar with basic probability theory (random variables, conditional expectation, convergence types). For the last month of lectures, some acquaintance with differential geometry will be useful, but not necessary, the required notions being briefly introduced.

Textbooks: Recommended texts are: Probability Essentials by Jacod-Protter for prerequisites, Multidimensional Diffusion Processes by Stroock-Varadhan and Continuous martingales and Brownian motion by Revuz-Yor.

Grading: problem sets (50%) and a final project (50%).

The course covers the following topics:

• Discrete martingales: stopping times, optional stopping theorem, maximal inequalities.
• Gaussian processes: Gaussian vectors, Brownian motion and some of its (ir)regularities properties, Kolmogorov's continuity criterion.
• Semimartingales: filtrations, local martingales.
• The stochastic integral, Itô's formula and applications: Girsanov's theorem, the Dirichlet problem, recurrence and transience of multidimensional Brownian motion, conformal invariance.
• Stochastic differential equations: type of solution, the Lipschitz case, piecewise approximations, shifts in the Cameron-Martin space.
• Representations: Itô's representation, the chaos decomposition, the Ohrenstein-Uhlenbeck semigroup.
• Differential calculus: the Gross-Sobolev derivative, the Clark-Ocone formula.
• Concentration of measure: hypercontractivity, logarithmic Sobolev inequalities, the Bakry-Emery curvature criterium, concentration on discrete spaces (the Kahn-Kalai-Linial theorem).
• Trace formulae: the Bismut approach in simple cases (Poisson, Selberg formulae)?

• Problem set 1.
• Problem set 2.
• Problem set 3.
• Problem set 4.
• Problem set 5.

• December 3rd 2010, 11am, Ricci curvature and (non) explosion of Brownian motion, Stephan Zheng.
• December 6th 2010, 2pm, The Anderson model, the DMPK equation and random matrices, Maximilian Butz.
• December 8th 2010, 10am, The Dirichlet process, Tatsu Hashimoto.
• December 8th 2010, 11am, Local universality for eigenvalues of random matrices, Dana Mendelson.
• December 10th 2010, 10am, Concentration of measure and model selection, Albert Shieh.
• December 10th 2010, 11am, Hausdorff dimension and local time for Brownian motion, Jamin Sheriff.
• December 10th 2010, 2pm, Random surfaces, Plancherel measures, Jeffrey Kuan.
• December 10th 2010, 3pm, An Introduction to the Noisy Oscillator, Philip Owrutsky .