This EPSRC-funded project is aimed at investigating the limits of probabilistic coupling. See the books by Lindvall and Thorisson for good expositions of this most important technique, or look at the lecture notes of a course I have given at EPFL Lausanne for a personal view. I worked on this with my EPSRC-funded research associate Sayan Banerjee over the period November 2013 - June 2016. From June 2016 - September 2016, also funded by this grant, Giacomo Zanella worked with me on an associated problem related to Markov chain Monte Carlo and Dirichlet forms.
In initial work (Kendall, 2014a), I conducted a case study of one of the simplest possible cases, namely the question of coupling Brownian motion and its local time at zero. I established that in that case Markovian couplings could not be efficient, though there was a simple reflection/synchrnous coupling which was optimal in the class of all Markovian couplings. In this paper I also managed to use this model case to establish a coupling result for the BKR diffusion, which relates to the Brownian nature of the BKR filtration.
Sayan and I established (in Banerjee & Kendall, 2016a) that maximal probabilistic couplings for smooth elliptic diffusions can only be Markovian in very special cases (state-space has to be simply connected and of constant curvature, drift has to derive from rotational symmetries or -- in the Euclidean case -- dilations). Previous work by Kuwada and by Hsu and Sturm had produced similar results for Brownian motion; we showed how to use some old Riemannian geometry results to obtain a rather general theorem for elliptic diffusions. This is a striking result that shows the importance of developing good understanding of non-maximal but otherwise efficient couplings.
We then continued (in Banerjee & Kendall, 2016b) by investigating probabilistic couplings for the Kolomogorov diffusion (the simplest nilpotent diffusion). We have shown that in this case the class of Markovian couplings cannot be efficient, but we have exhibited an efficient non-Markovian coupling. This case study adds significantly to the knowledge base of what kinds of couplings can be derived in various circumstances. The efficient non-Markovian construction is of particular interest: it exploits the infinite-dimensional Karhunen-Loeve representation of Brownian motion.
Developing the work on the Kolmogorov diffusion, we have now established how to couple finite sets of polynomial stochastic integrals for two-dimensional diffusions, using only reflection and synchronous couplings. This work is now being prepared for publication. While we have not yet attained our ultimate goal (to exhibit couplings for general nilpotent diffusions), we believe that the two-dimensional result sets the template for future work in which we plan together to conquer this full objective. (Banerjee et al, 2016, show how to use these ideas to obtain useful gradient estimates for the Heisenberg group.)
Work (myself in collaboration with Candellero) is also being prepared for publication concerning coupling in infinite-dimensional situations. This has links with the work on Kolmogorov diffusion coupling, and makes intriguing links with Banach-space geometry.
Other related work carried out in the grant period includes:
Connor & Kendall (2015) on perfect simulation (an important coupling technique) for multi-server queues. This was the first result on implementable perfect simulation for the general case of M/G/c queues; it has now been followed up by a variety of other works.
Ernst et al (2017) have established a general context for maximal (not necessarily Markovian) couplings in a reverse-time context (hence described as "un-coupling", maximal exit, or MEXIT) . The paper also expounds a variety of applications to statistics and particularly Markov chain Monte Carlo.
In Kendall (2014b, 2017), I produced work on a particular scale-invariant random spatial network. A useful exposition is given in Kendall (2014c).
Finally my work with Zanella together with Bédard (Zanella et al, 2016) shows how to use stochastic analysis methods to address optimal scaling for Markov chain Monte Carlo. This is likely to develop into research programme of its own.
Here is a list of papers which concern work funded wholly or in part by this grant:
- S. Banerjee, Gordina, M., & Mariano, P. (2016). Coupling in the Heisenberg group and its applications to gradient estimates. Submitted for publication. Available from arXiv:1610.06430.
- S. Banerjee & Kendall, W. S. (2016a). Rigidity for Markovian maximal couplings of elliptic diffusions. Probability Theory and Related Fields, 42(to appear), 58pp. Gold Access: http://doi.org/10.1007/s00440-016-0706-4.
- S. Banerjee & Kendall, W. S. (2016b). Coupling the Kolmogorov Diffusion: maximality and efficiency considerations. Advances in Applied Probability, 48(A), 15–35. Available from arXiv:1506.04804.
- S. B. Connor & W.S. Kendall (2015), Perfect Simulation of M/G/c Queues. Advances in Applied Probability, 47(4), 1039–1063. Available from arXiv:1402.7248.
- P. Ernst, Kendall, W. S., Roberts, G. O., & Rosenthal, J. S. (2017). MEXIT: Maximal uncoupling times for Markov processes. Submitted for publication. Available from arXiv:1702.03917.
- W. S. Kendall (2014a), Coupling, local times, immersions. The Bernoulli Journal 21(2), 1014–1046. Available from arXiv:1212.1670.
- W. S. Kendall (2014b). Return to the Poissonian city. Journal of Applied Probability, 51A(A), 297–309. Available from arXiv:1309.7645.
- W. S. Kendall (2014c). Lines and networks. Markov Processes and Related Fields, 20(1), 81–106.
- W. S. Kendall (2017). From Random Lines to Metric Spaces. Annals of Probability 45(1), 469–517. Available from arXiv:1403.1156.
- G. Zanella, Kendall, W. S., & Bédard, M. (2016). A Dirichlet Form approach to MCMC Optimal Scaling. To appear in Stochastic Processes and Their Applications. Available from arXiv:1606.01528.
In conclusion, here is a mindmap generated in the early stages of the project.