Probabilistic coupling and nilpotent diffusions

现代概率论的一个主要主题是，通过比较基于不同但相互关联的随机性来源的系统的两个副本的行为，人们可以发现关于随机系统的许多信息。最简单的例子是对称随机游走，就像在公平的抛硬币游戏中出现的那样。随机漫步以等概率独立向上或向下移动。考虑两个这样的随机漫步，从时间0开始，高度分别为-k和k。假设它们的随机性是通过反射相互关联的，一个向上移动另一个向下移动，反之亦然。非常简单的论证表明，这两种行为最终几乎肯定会相遇（“成对”）；当它们第一次（同时）访问高度0时，它们就会这样做。这种所谓的反射耦合可以被广泛地推广：更一般的随机漫步，相互作用的粒子系统，使用一种称为路径耦合的技术的许多种离散随机系统，以及使用随机演算的连续随机系统。事实证明，该方法为分析这些系统提供了强大的技术。例如，收敛到随机平衡的速度（在许多应用中是一个关键问题）是由耦合发生的速度控制的。这个项目是对这个问题的概括。我们能不能不仅耦合系统，而且同时耦合系统的一些函数？在随机漫步的例子中，有人可能会问，我们是否不仅可以耦合随机漫步，还可以耦合它们轨迹下的（带符号的）区域？知道这个问题的一般答案将大大增加使用耦合技术分析随机系统的可能性。困难在于功能不受系统本身所受的那种直接控制。随机游动的跳跃可以直接相关或反相关。轨迹下的面积只能间接受到影响。出现的问题非常类似于当一个人试图把车停在一个狭窄的停车位时发生的问题：一个人想要把车移到一边（类似于直接控制轨迹下的区域），但只能改变汽车的前后运动，以及一些方向上的轻微改变（类似于控制跳跃之间的相关性）。我们现在知道很多情况下，答案是我们可以把函数结合起来。例如，以连续系统为例，作为一个清晰的技术案例，我们现在知道如何控制轨迹下的区域。这个项目的目的是将其扩展到最一般的情况，其中答案可能是肯定的，至少在连续系统的情况下：即所谓的幂零扩散的情况。先验地说，这似乎很有野心；有人可能会认为，更有可能的是，控制隐式额外功能集的选项非常有限。但现在看来，答案很可能是肯定的，基于一些已经建立了耦合的关键例子，以及在这样做时采用的技术，我们已经能够制定一个程序，通过这个程序可以找到证明。这个项目是关于证明一般结果，估计产生耦合的速率，并将结果与其他数学领域和最佳运输（如何有效地将大量材料从一组位置移动到另一组位置）中的应用，以及统计模拟（在随机计算机算法和现代统计估计的研究中很重要），随机动力系统（例如在全球气象学和海洋动力学中出现）和粗糙路径理论。

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.