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This module is taught as part of the MIGSAA CDT programme and offered across SMSTC.

If you want to take this course, please register straight away otherwise we will not know which video conference sites are needed. Staff and students should log into the SMSTC site, click on My Account and then My SMSTC and follow the instructions there.

**VENUE** (for those who can attend in person): TBA

The theory of diffusion processes has a very rich mathematical structure. One of the key features of such a theory is the interplay between probabilistic and analytic techniques. Analytic techniques are employed to give a macroscopic description of the dynamics, while probabilistic tools (stochastic analysis and stochastic processes) are used for the microscopic description.

Course duration and format: 20 hours (correspondingly, 15 credits); the first half of the course will consist of taught lectures while the second part will be organised completely reading-group style.

The course will present the theory of time-homogeneous diffusion processes from the analysis standpoint. It will be split in two parts of roughly 10 hours each.

Students who have attended the course in 2015 or in 2018 should skip the first part but they are very welcome to attend the second part

(and in this case they would be awarded 7.5 credits).

The first half of the course will consist of taught lectures; covered topics will include:

- Markov Semigroups and their generators. Dual semigroup. Invariant and reversible measures
- Ergodic Theory for Markov Semigroups: Strong Feller semigroups, Krylov-Bogoliubov Theorem, Doob's Theorem, Prokhorov's theorem
- Backward Kolmogorov and Fokker-Planck equation
- Reversible diffusions: spectral gap inequality, exponentially fast convergence toequilibrium
- Over and under-dumped Langevin equation

The second half of the course will be reading group style, no taught lectures at all.

Every week a student will cover a different topic - or, if the students prefer and their taste is homogeneous, we could embark in the detailed reading of material on just one topic. Precise choice of articles/ book sections to read will be decided depending on the interests of the audience.

- Possible topics could include:
- Hypoelliptic diffusions and the Hoermander condition: analytical, probabilistic

and geometrical aspects; - Important examples of diusions in statistical mechanics: heat bath models

and second order Langevin equation - Non-reversible diffusions
- Hypocoercivity Theory
- Diffusion processes for sampling
- Time-inhomogeneous diffusions
- Interacting Particle systems

Lecture notes will be provided, containing an extensive bibliography. The lecture notes will be complemented with further reading material. An indicative (non-final) list of references is the following

- Analysis and Geometry of Markov Diffusion operators, by D. Bakry et al.
- Second Order Partial Differential Equations in Hilbert Spaces, Da Prato and

Zabczyk - To begin at the beginning... D. Williams
- Analytical Methods for Markov Semigroups. L. Lorenzi and M. Bertoldi
- Hypoelliptic Second order differential equations. L. Hoermander
- On Malliavin's proof of Hoermander's theorem. M. Hairer
- How hot can a heat bath get? M. Hairer

Further and more detailed references will be given during the course.

For the first part of the course: I will distribute some questions that the students can solve at home. For the second part: assessment will come from

giving presentations and from active participation.

The background for this course is given by the Probability 1 SMSTC stream (and some lectures of the Probability 2 stream), where basic stochastic calculus and the basic theory of Markov Processes is covered.

Moreover students with a background in analysis will nd obvious relations with the theory of parabolic PDEs.

There are strong conceptual links with the courses on dissipative PDEs as well.

Basic probability theory, basic stochastic calculus (e.g. Ito formula), very basic SDE and PDE theory.

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Mail : johanna.mcbryde@icms.org.uk