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Faculty of Mathematics

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Course tutor: Professor Nathanael Berestycki and Dr Richard Nickl

This course covers the most important ways that randomness is incorporated into mathematical models and then investigates how these models are analysed and their behaviour understood.

The first key notion is that of a Markov process: the basic theory of Markov chains and Brownian motion is briefly reviewed and directed reading provided for any students needing to reinforce understanding in these areas. Limit techniques for random processes are surveyed, including fluid limits, diffusion approximation, large deviations, homogenization. Next, a range of examples is identified, from applications, for example in stochastic networks, genetics or population processes. Student teams will prepare and present asymptotic analyses of assigned examples.

The last part of the course deals with stochastic particle models: topics might include scaling limits and nonlinear evolution equations, high-dimensional simulation and filtering, mixing times and coupling from the past.