M. Scheutzow (TU Berlin), A. Kulik (Uni Kiev/TU Berlin)
M. Scheutzow (TU Berlin), Coupling, transportation plans and Markov chains
Coupling has become a powerful tool in probability theory during the past few decades. It allows to compare random variables which are a priori unrelated. Coupling has been applied in a broad variety of contexts, e.g. to prove limit theorems, to derive inequalities, or to obtain approximations. We will introduce the term in the lecture and provide some basic examples. In particular, we will see that couplings and "transportation plans" are essentially the same objects. We will then show how couplings can be used to prove uniqueness of and convergence to an invariant probability measure for a Markov chain with a very elementary and explicit coupling approach, this approach to "Doob's theorem" being the highlight of the talk. The talk aims at an audience with little knowledge in probability theory: knowing the concept of a "measure" and of a "(Markov) kernel" or "Markov chain" will be more than sufficient. Everything else will be explained. This last part of the talk is based on joint work with Alexei Kulik (Kiev).
A. Kulik (Uni Kiev/TU Berlin), Generalized couplings and Markov chains
The talk will be devoted to the notion of "generalized coupling'', which is a natural extension of "coupling'' introduced in the previous talk. Generalized coupling are well-suited to handle realistic Markov models with complicated state spaces, e.g. those described by stochastic delay equations and stochastic PDEs, two examples which will be discussed in the talk. The crux of a usual "coupling'' approach, the construction of the required coupling, is drastically simplified in the "generalized coupling'' framework, which makes the method very efficient. The talk aims at an audience with basic knowledge in probability theory and differential equations. The talk is based on a joint work with Michael Scheutzow.