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Probability: Math 418/544 and 419/545
This pair of courses provides a thorough introduction to measure-theoretic probability. Prior knowledge of measure theory or probability is not assumed. Results from measure theory are developed and used as needed. The focus is on discrete time and continuous time stochastic processes. Topics include: martingales, law of large numbers, central limit theorem, Brownian motion and special topics.
Stochastic Analysis: Math 546
This is a rigorous course on finite-dimensional continuous stochastic processes, focusing on Markov processes. Topics include: stochastic integration with respect to continuous semimartingales, Itô's formula for continuous semimartingales and applications, stochastic differential equations, Girsanov's formula, martingale problems.
Additional topics depending on the interests of the class may then be chosen from: one-dimensional diffusion theory, local time, introduction to SLE, applications to areas such as filtering, stochastic control, genetics, mathematical finance, Stroock-Varadhan theory for finite dimensional diffusions.
Prerequisites: Math 545 or consent of the instructor. Students from other Departments interested in learning about stochastic analysis from a mathematical perspective are encouraged.
Discrete Probability: Math 548
This course covers more advanced topics in discrete probability. Some probability background is needed, including Markov chains and martingales. Measure theory may be used at some points. Topics include spectral analysis of Markov chains and mixing times, electrical networks and random walks, random graphs (Erdős-Rényi, random regular graphs, etc.), percolation (leading up to Smirnov's theorem on conformal invariance) and other statistical mechanics models (Ising, Potts).
Topics in Probability: Math 608
This is a topics course in probability which is offered when there is sufficient student interest. The topic of the course changes from year to year depending on the interests of students and instructor.
Topics in Mathematical Physics: Math 609
This topics course often studies a subject of interest to graduate students in probability.