Probabilistic coupling and nilpotent diffusions

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A major theme of modern probability is that one can discover much about a random system by comparing the behaviour of two copies of the system, based on different but inter-related sources of randomness. The simplest example is that of a symmetric random walk, as might arise in a fair coin-tossing game. The random walk moves independently up or down with equal probability. Consider two such random walks, begun at time 0 at heights -k and k respectively. Suppose their randomness is interrelated by reflection, so that one moves up as the other moves down, and vice versa. Very simple arguments show that the two walks must almost surely eventually meet ("couple"); and they will do so when they first (and simultaneously) visit height 0.This so-called reflection coupling can be vastly generalized: to more general kinds of random walks, to interacting particle systems, to many sorts of discrete random system using a technique called path-coupling, and to continuous random systems using stochastic calculus. It turns out that the method delivers powerful techniques for analyzing these systems. For example, rate of convergence to stochastic equilibrium (a crucial question in many applications) is controlled by the rate at which coupling occurs.The project concerns a generalization of this question. Can we couple not only the system, but also and simultaneously some functionals of the system? In the random walk example, one might ask whether we can couple not only the random walks, but also the (signed) areas under their trajectories? Knowing a general answer to this question would substantially increase the possibilities for using the coupling technique to analyze random systems. The difficulty is that the functional is not subject to the same kind of direct control as is the system itself. The jumps of the random walks can be correlated or anti-correlated directly. The areas under the trajectories can only be affected indirectly. The problems that arise are very similar to those that occur when one tries to park a car in a confined parking slot: one would like to move the car sideways (analogous to directly controlling the areas under the trajectories), but can only alter the forwards-and-backwards motion of the car, together with some slight changes in direction (analogous to controlling the correlation between the jumps).We now know a number of cases in which the answer is that we can couple functionals. For example, taking the case of continuous systems as a clean technical case, we now know how to control areas under trajectories. The aim of this project is to extend this to cover the most general possible case in which the answer might be expected to be yes, at least in the case of continuous systems: namely the case of so-called nilpotent diffusions. A priori this seems very ambitious; one might suppose it more likely that the options to control implicitly sets of extra functionals are very limited. But it now seems very likely that the answer is yes, based on a number of key examples in which coupling has been established, and based on the techniques adopted when doing this, and we have been able to set down a programme by which a proof may be found. This project is about proving the general result, estimating the rate at which the resulting coupling will occur, and relating the result both to other areas of mathematics and to applications in optimal transportation (how to move volumes of material efficiently from one set of locations to another), to statistical simulation (important in the study of randomized computer algorithms and in modern statistical estimation), stochastic dynamical systems (as arise for example in global meteorology and ocean dynamics), and the theory of rough paths.