The Annals of Applied Probability

Efficient Markovian couplings: examples and counterexamples

Krzysztof Burdzy and Wilfrid S. Kendall

Full-text: Open access


In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of “perfect simulation”: it helps to establish the “price of perfection” for such methods. In general, one can always achieve efficient coupling if the coupling is allowed to “cheat”(if each component’s behavior is affected by the future behavior of the other component), but the situation is more interesting if the coupling is required to be co-adapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains.

Article information

Ann. Appl. Probab. Volume 10, Number 2 (2000), 362-409.

First available in Project Euclid: 22 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60H30: Applications of stochastic analysis (to PDE, etc.) 65U05

Diffusion Chen-optimal coupling co-adapted coupling couplling exponent efficient coupling efficient coupling heuristic exact simulation Markov chain mirror coupling monotonicity perfect simulation price of perfection reflecting Brownian motion spectral gap synchronous coupling


Burdzy, Krzysztof; Kendall, Wilfrid S. Efficient Markovian couplings: examples and counterexamples. Ann. Appl. Probab. 10 (2000), no. 2, 362--409. doi:10.1214/aoap/1019487348.

Export citation


  • [1] Aldous, D. and Fill, J. A. (1996). Reversible Markov chains and random walks on graphs. Unpublished manuscript.
  • [2] Aldous, D. J. and Diaconis, P. (1987). Strong uniform times and finite random walk. Adv. in Appl. Math. 8 69-97.
  • [3] Ba nuelos, R. and Burdzy, K. (1999). On the "hot spots" conjecture of J. Rauch. J. Funct. Anal. 164 1-33.
  • [4] Bass, R. and Burdzy, K. (1992). Lifetimes of conditioned diffusions. Probab. Theory Related Fields 91 405-443.
  • [5] Bass, R. and Burdzy, K. (1999). Fiber Brownian motion and the "hot spots" problem. Duke Math. J. To appear.
  • [6] Burdzy, K. and Werner, W. (1999). A counterexample to the "hot spots" conjecture. Ann. Math. 149 309-317.
  • [7] Cai, H. (1997). A note on an exact sampling algorithm and Metropolis Markov chains. Technical report, Univ. Missouri, St. Louis.
  • [8] Chen, M.-F. (1992). From Markov Chains to Non-equilibrium Particle Systems. World Scientific, Singapore.
  • [9] Chen, M.-F. (1994). Optimal Markovian couplings. Progr. Natur. Sci. (English Ed.) 4 364- 366.
  • [10] Chen, M.-F. (1994). Optimal Markovian couplings and applications. Acta Math. Sinica (N.S.) 10 260-275.
  • [11] Chen, M.-F. (1998). Trilogy of couplings and general formulas for lower bound of spectral gap. In Proceedings of the Symposium on Probability Towards the Year 2000 (L. Accardi and C. Heyde, eds.) 123-136. Springer, New York.
  • [12] Chen, M. -F. and Wang, F.-Y. (1994). Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37 1-14.
  • [13] Chung, K. L. (1984). The lifetime of conditioned Brownian motion in the plane. Ann. Inst. H. Poincar´e Probab. Statist. 20 349-351.
  • [14] Cox, D. R. and Miller, H. D. (1965). The Theory of Stochastic Processes. Methuen, London.
  • [15] Cranston, M. and Le Jan, Y. (1989). On the noncoalescence of a two point Brownian motion reflecting on a circle. Ann. Inst. H. Poincar´e Probab. Statist. 25 99-107.
  • [16] Cranston, M. and Le Jan, Y. (1990). Noncoalescence for the Skorohod equation in a convex domain of 2. Probab. Theory Related Fields 87 241-252.
  • [17] Diaconis, P. and Fill, J. (1990). Strong stationary times via a new form of duality. Ann. Probab. 18 1483-1522.
  • [18] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750.
  • [19] Dobru sin, R. (1971). Markov processes with a large number of locally interacting components (in Russian). Problemy Peredachi Informatsii 7 70-87.
  • [20] Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York.
  • [21] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth, Pacific Grove, CA.
  • [22] Fill, J. (1998). An interruptible algorithm for exact sampling via Markov chains. Ann. Appl. Probab. 8 131-162.
  • [23] Folland, G. B. (1976). Introduction to Partial Differential Equations. Princeton Univ. Press.
  • [24] Griffeath, D. (1975). A maximal coupling for Markov chains. Z. Wahrsch. Verw. Gebiete 31 95-106.
  • [25] Griffeath, D. (1978). Coupling methods for Markov processes. In Studies in Probability and Ergodic Theory, Advances in Mathematics, Supplementary Studies 2 1-43. Academic, New York.
  • [26] Ingrassia, S. (1993). Geometric approaches to the estimation of the spectral gap of reversible Markov chains. Combin. Probab. Comput. 2 301-323.
  • [27] It o, K. and McKean, H. P. (1974). Diffusion Processes and Their Sample Paths, 2nd ed. Springer, New York.
  • [28] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.
  • [29] Kendall, W. S. (1989). Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel. J. Funct. Anal. 86 226-236.
  • [30] Kendall, W. S. (1998). Stochastic calculus in Mathematica: software and examples. Technical Report 333, Dept. Statistics, Univ. Warwick.
  • [31] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • [32] Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
  • [33] Liu, J. S. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statist. Comput. 6 113-119.
  • [34] Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Probab. 6 218-237.
  • [35] Matthews, P. (1992). Strong stationary times and eigenvalues. J. Appl. Probab. 29 228-233.
  • [36] Meise, Ch. (1996). Lower bounds for the spectral gap of Markov chains. Preprintreihe Sonderforschungsbereich 343 "Diskrete Strukturen in der Mathematik" 96-019, Univ. Bielefeld.
  • [37] Mountford, T. S. and Cranston, M. (1999). Efficient coupling on the circle. Unpublished manuscript.
  • [38] Pinsky, M. A. (1980). The eigenvalues of an equilateral triangle. SIAM J. Math. 11 819-827.
  • [39] Pinsky, M. A. (1985). Completeness of the eigenfunctions of the equilateral triangle. SIAM J. Math. Anal. 16 848-851.
  • [40] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223-252.
  • [41] Roberts, G. O. and Rosenthal, J. S. (1999). Convergence of slice sampler Markov chains. J. Roy. Statist. Soc. Ser. B 61 643-660.
  • [42] Rosenthal, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Statist. Comput. 6 269-275.
  • [43] Saloff-Coste, L. (1998). Simple examples of the use of Nash inequalities for finite Markov chains. In Stochastic Geometry: Likelihood and Computation (O. E. BarndorffNielsen, W. S. Kendall and M. N. M. van Lieshout, eds.) 365-400. Chapman and Hall, London.
  • [44] Sharpe, M. (1989). General Theory of Markov Processes. Academic, New York.
  • [45] Smith, R. L. and Tierney, L. (1996). Exact transition probabilities for the Independence sampler. Technical report, Univ. North Carolina.
  • [46] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701-1728.
  • [47] Varadhan, S. R. S. and Williams, R. J. (1985). Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405-443.
  • [48] Wang, F.-Y. (1994). Application of coupling methods to the Neumann eigenvalue problem. Probab. Theory Related Fields 98 299-306.