Skip to main content Accessibility help

Coupling the Kolmogorov diffusion: maximality and efficiency considerations

  • Sayan Banerjee (a1) and Wilfrid S. Kendall (a1)


This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral or, more generally, together with a finite number of iterated time integrals). We show that in this case there can be no Markovian maximal coupling. Indeed, there can be no efficient Markovian coupling strategy (efficient for all pairs of distinct starting values), where the notion of efficiency extends the terminology of Burdzy and Kendall (2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling strategy.


Corresponding author

Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address:
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address:


Hide All
[1] Alili, L. and Wu, C. T. (2014).Müntz linear transforms of Brownian motion.Electron. J. Prob. 19, 15pp.
[2] Aue, A.,Horváth, L. and Hus̆ková, M. (2009).Extreme value theory for stochastic integrals of Legendre polynomials.J. Multivariate Anal. 100,10291043.
[3] Bailleul, I. (2008).Poisson boundary of a relativistic diffusion.Prob. Theory Relat. Fields 141,283329.
[4] Banerjee, S. and Kendall, W. S. (2014).Rigidity for Markovian maximal couplings of elliptic diffusions. Preprint. Available at
[5] Ben Arous, G.,Cranston, M. and Kendall, W. S. (1995).Coupling constructions for hypoelliptic diffusions: two examples. In Stochastic Analysis (Ithaca, NY, 1993; Proc. Sympos. Pure Math. 57), eds M. C. Cranston and M. A. Pinsky,American Mathematical Society,Providence, RI, pp. 193212.
[6] Burdzy, K. and Kendall, W. S. (2000).Efficient Markovian couplings: examples and counterexamples.Ann. Appl. Prob. 10,362409.
[7] Da Prato, G. and Zabczyk, J. (2014).Stochastic Equations in Infinite Dimensions (Encyclopedia Math. Appl. 152),2nd edn.Cambridge University Press.
[8] Doeblin, W. (1938).Exposé de la théorie des chaînes simples constantes de Markoff à un nombre fini d'états.Revue Math. de l'Union Interbalkanique 2,77105.
[9] Goldstein, S. (1979).Maximal coupling.Prob. Theory Relat. Fields 46,193204.
[10] Griffeath, D. (1975).A maximal coupling for Markov chains.Z. Wahrscheinlichkeitsth. 31,95106.
[11] Groeneboom, P.,Jongbloed, G. and Wellner, J. A. (1999).Integrated Brownian motion, conditioned to be positive.Ann. Prob. 27,12831303.
[12] Jansons, K. M. and Metcalfe, P. D. (2007).Optimally coupling the Kolmogorov diffusion, and related optimal control problems.LMS J. Comput. Math. 10,120.
[13] Kearney, M. J. and Majumdar, S. N. (2005).On the area under a continuous time Brownian motion till its first-passage time.J. Phys. A 38,40974104.
[14] Kendall, W. S. (2007).Coupling all the Lévy stochastic areas of multidimensional Brownian motion.Ann. Prob. 35,935953.
[15] Kendall, W. S. (2010).Coupling time distribution asymptotics for some couplings of the Lévy stochastic area. In Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman (London Math. Soc. Lecture Note Ser. 378), eds N. H. Bingham and C. M. Goldie,Cambridge University Press, pp. 446463.
[16] Kendall, W. S. (2015).Coupling, local times, immersions.Bernoulli 21,10141046.
[17] Kendall, W. S. and Price, C. J. (2004).Coupling iterated Kolmogorov diffusions.Electron. J. Prob. 9,382410.
[18] Kunita, H. (1997).Stochastic Flows and Stochastic Differential Equations (Camb. Stud. Adv. Math. 24).Cambridge University Press.
[19] Kuwada, K. (2009).Characterization of maximal Markovian couplings for diffusion processes.Electron. J. Prob. 14,633662.
[20] McKean, H. P., Jr. (1963).A winding problem for a resonator driven by a white noise.J. Math. Kyoto Univ. 2,227235.
[21] Pitman, J. W. (1976).On coupling of Markov chains.Z. Wahrscheinlichkeitsth. 35,315322.
[22] Rosenthal, J. S. (1997).Faithful couplings of Markov chains: now equals forever.Adv. Appl. Math. 18,372381.
[23] Smith, A. M. (2013).A Gibbs sampler on the n-simplex.Ann. Appl. Prob. 24,114130.
[24] Sverchkov, M. Yu. and Smirnov, S. N. (1990).Maximal coupling for processes in D[0,∞].Dokl. Akad. Nauk SSSR 311,10591061 (in Russian). English translation:Soviet Math. Dokl. 41 (1990),352354.
[25] Thorisson, H. (1994).Shift-coupling in continuous time.Prob. Theory Relat. Fields 99,477483.
[26] Wellner, J. A. and Smythe, R. T. (2002).Computing the covariance of two Brownian area integrals.Statist. Neerlandica 56,101109.
[27] Yan, L. (2004).Two inequalities for iterated stochastic integrals.Arch. Math. (Basel) 82,377384.


MSC classification

Coupling the Kolmogorov diffusion: maximality and efficiency considerations

  • Sayan Banerjee (a1) and Wilfrid S. Kendall (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed