### Talk by WSK at London Probability Seminar (Imperial College), 28 October 2013

This talk will be in the afternoon. In the morning Hermann Thorisson will be giving 2 lectures in the morning to introduce probabilistic coupling, as well as giving a seminar in the afternoon.

My own talk title and abstract:

**Title:**

Probabilistic coupling and Nilpotent Diffusions

**Abstract:**

Modern probability theory makes considerable use of the technique of *probabilistic coupling*. The idea is, to analyse a given random process by constructing two inter-dependent copies of it, defined on the same probability space, and related in such a way as to facilitate analysis. Applications include: establishing monotonicity in non-obvious situations, developing quantitative approximations to distributions of random variables, constructing gradient estimates, and even producing exact simulation algorithms for Markov chains.

However the thematic question, which has driven much of the theory of probabilistic coupling, concerns whether or not one can construct the two coupled random processes so that they almost surely meet ("couple") at some future random time, and if so then whether one can construct a maximal coupling, for which the random time is smallest possible? The question is sharpened if we require the coupling to be *co-adapted* (also: immersed, or Markovian); this is an additional requirement that the coupling respect the underlying causal structure of the random processes, and can be viewed as implying that the coupling is easily constructable in some general sense.

There is a considerable body of theory describing how to build successful co-adapted couplings for elliptic diffusions, all building on the basic *reflection coupling* for simple random walks or Brownian motion (very simply, the random jumps of the coupled process are arranged so far as possible to be the opposites of the random jumps of the original process). It is conjectured that successful co-adapted couplings can be built for all hypoelliptic diffusions (diffusions in d dimensions with fewer than d "directions of randomness").

In this talk I will survey the general theory of coupling, describe the known results for co-adapted couplings of hypoelliptic diffusions (in fact, Brownian motions on nilpotent Lie groups), and briefly discuss a related and very simple example which has applications to the theory of filtrations.

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