STOR Colloquium: Sayan Banerjee, University of Warwick

Couplings and geometry

A coupling of (the laws of) two Markov processes specifies a particular construction of copies of the two processes simultaneously on the same space. They have a long history and find numerous applications in probability and analysis, ranging from yielding bounds on the mixing times of Markov chains to studying harmonic maps.

It is natural to ask whether we can construct a coupling where the coupled processes actually meet (successful coupling). If such a coupling exists, how fast can we make them meet (coupling rate)? It turns out that this question has deep connections with the generator of the Markov process and the geometry of the underlying space.

In this talk, I will give an overview of some results in this area. In particular, we will focus on general elliptic diffusions on Riemannian manifolds, and show how geometry (dimension of the isometry group, flows of isometries, Killing vector fields and dilation vector fields) plays a fundamental role in relating the space and the generator of the diffusion to the coupling rate. I will also briefly describe efficient coupling techniques for some nilpotent diffusions using ideas from the theory of infinite dimensional Brownian motion.

This is joint work with W.S. Kendall.