# Foundations of probabilistic coupling

I have a long-standing interest in probabilistic coupling (the art of construction of two interdependent copies of a random process in such a manner as to facilitate a specific argument (perhaps concerning approximation, or convergence, or monotonocity). My webpages contain a copy of slides from a set of survey lectures which explain more.

A principal question is, if I can couple two copies of a diffusion to meet at a random coupling time, then when can I do the same if it is required simultaneously to couple suitable path functionals (examples: time integrals, stochastic areas). To make this interesting (and to make calculations feasible), we restrict to Markovian couplings (also called immersion or co-adapted couplings): the coupling interdependence has to respect the causal structure underlying the construction of the individual random processes. Substantial progress has been made (Ben Arous et al, 1995; Kendall & Price, 2004; Kendall, 2007, 2010, 2014), and we are now hot on the trail of a big general result, but there are many intriguing side-questions. For example, one might ask about coupling in infinite-dimensional problems (Candellero and Kendall, 2017), or wonder how fast such couplings might occur (Banerjee and Kendall 2016b): this has relevance to important questions about gradient estimates in mathematical analysis.

Recent work has included rather definitive answers to the question, when can maximal couplings be Markovian? (Banerjee and Kendall, 2016a) However there remains a great deal to be done. Getting good insights will result in major advances in our understanding of this probabilistic technique.

1. Banerjee, S., & Kendall, W. S. (2017). Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case. arXiv, 1705.01600, 45pp. See arXiv, 1705.01600.
2. Banerjee, S., & Kendall, W. S. (2016a). Rigidity for Markovian Maximal Couplings of Elliptic Diffusions. Probab. Theory Related Fields 42, 58pp. See also arXiv, 1412.2647.
3. Banerjee, S., & Kendall, W. S. (2016b). Coupling the Kolmogorov Diffusion: maximality and efficiency considerations. Advances in Applied Probability 48A, 15-35. See also arXiv, 1506.04804.
4. Ben Arous, G., Cranston, M., & Kendall, W. S. (1995). Coupling constructions for hypoelliptic diffusions: Two examples. In M. Cranston & M. Pinsky (Eds.), Stochastic Analysis: Proceedings of Symposia in Pure Mathematics 57, 193–212. Providence, RI Providence: American Mathematical Society.
5. Candellero, E., & Kendall, W. S. (2017). Coupling of Brownian motions in Banach spaces. arXiv, 1705.08300, 12 pp. See arXiv, 1705.08300.
6. Kendall, W. S. (2007). Coupling all the Lévy stochastic areas of multidimensional Brownian motion. The Annals of Probability, 35(3), 935–953.
7. Kendall, W. S. (2010). Coupling time distribution asymptotics for some couplings of the Lévy stochastic area. In N. H. Bingham & C. M. Goldie (Eds.), Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman (pp. 446–463). Cambridge: Cambridge University Press.
8. Kendall, W. S. (2015). Coupling, local times, immersions. Bernoulli, 21.2, 1014-1046.
9. Kendall, W. S., & Price, C. J. (2004). Coupling iterated Kolmogorov diffusions. Electronic Journal of Probability, 9(Paper 13), 382–410.