What is it about?
Coupling is a major tool in probability which allows us to gain elegant and deep understanding of random systems by constructing coupled realizations: copies of the system which are not independent but related in some insightful way. The thematic way to do this is to couple random processes so that they meet at some future time, preferably as soon as possible. This already is very helpful: for example bounds on the rate of coupling translate directly into bounds on the speed at which statistical equilibrium is attained. Surprisingly, it is also possible to couple not only random processes but also selected functionals of these processes, so that not just the random processes but also the functionals agree at some random time. Examples include: real Brownian motion plus time integral, two-dimensional Brownian motion plus stochastic area, real Brownian motion plus local time at zero. In this paper we discuss that rate at which coupling occurs for real Brownian motion plus a finite set of iterated time integrals. Here we must make a distinction between Markovian coupling (the coupling respects the causal structure of the random process) and more general couplings (which need not do this). The paper shows that Markovian couplings will definitely happen at a slower rate than the best general couplings, but that finite-look-ahead couplings (violating causality only by looking ahead by a finite albeit increasing amount of time) can at least have rate of coupling asymptotically comparable to the best possible rate. The finite-look-ahead coupling works with an intriguing infinite-dimensional representation of the real Brownian motion using the Karhunen-Loeve infinite series expansion of Brownian motion.
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Why is it important?
This work is usefui as a case study to help us understand what we have to sacrifice when we respect the causal structure of the random processes we need to couple, and thus have to use Markovian coupling. It is part of a much larger programme of work aimed at establishing the limits and character of probabilistic coupling theory.
Perspectives
Personally I find the theory of probabilistic coupling absolutely fascinating. There is a kind of magic in figuring out how to make coupling happen, especially when one takes up the challenge of coupling not just the process but also associated functionals. So far the case history indicates that one can add in a surprisingly large number of possible functionals, so long as one is subtle about the coupling deployed. It is also becoming apparent that there are remarkable geometric connections: some cases when Markovian coupling is the best possible coupling (not withstanding causality restrictions) exhibit rather remarkable geometric structure. Both these questions are currently the subject of active research, and we expect some remarkable results over the next few years.
Wilfrid Kendall
University of Warwick
Read the Original
This page is a summary of: Coupling the Kolmogorov diffusion: maximality and efficiency considerations, Advances in Applied Probability, July 2016, Cambridge University Press,
DOI: 10.1017/apr.2016.40.
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