course home page: 5016 Topics in probability

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We will talk about several techniques used to study diffusions on curved spaces. First part
will be on what might be called geometry of diffusion operators. In particular, we will discuss
classical results such as Bochner-Weitzenbock formula, as well as more recent developments
such as curvature-dimension inequalities and related results due to Bakry, Emery, Ledoux et
al. The focus of this part of the course is proving functional inequalities using the geometry
of diffusion operators. This also leads to the heat kernel analysis on Riemannian manifolds
whose Ricci curvature is bounded from below. Some of these topics (along with many others)
are covered in a recent monograph
* Analysis and geometry of Markov diffusion operators * by D. Bakry, I. Gentil, and M. Ledoux. One of the possibilities after this is to study recent results in sub-Riemannian geometry with applications to hypoelliptic operators.

The second part of the course will deal with another technique called the coupling method. In particular, we will see how this method can be used in analysis to prove estimates such as gradient estimates etc. We will start with simple examples and then move to more geometric settings. If time permits we will talk about the Lindvall-Rogers and Cranston-Kendall couplings and recent developments for hypoelliptic diffusions.

The course assumes knowing basic real analysis, with no differential geometry background.

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Topics covered in the course by Masha Gordina

Masha Gordina (maria.gordina at uconn.edu)