Coupling Iterated Kolmogorov Diffusions

Wilfrid S Kendall (University of Warwick)
Catherine J. Price (Lehman Brothers)


The Kolmogorov-1934 diffusion is the two-dimensional diffusion generated by real Brownian motion and its time integral. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for Brownian motion, its time integral, and its twice-iterated time integral; and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 382-410

Publication Date: April 29, 2004

DOI: 10.1214/EJP.v9-201


  1. Aldous, David J.; Thorisson, Hermann. Shift-coupling. Stochastic Process. Appl. 44 (1993), no. 1, 1-14. MR1198659 (94f:60066)

  2. Ben Arous, Gérard; Cranston, Michael; Kendall, Wilfrid S. Coupling constructions for hypoelliptic diffusions: two examples. Stochastic analysis (Ithaca, NY, 1993), 193-212, Proc. Sympos. Pure Math., 57, Amer. Math. Soc., Providence, RI, 1995. MR1335472 (96d:60124)

  3. Burdzy, Krzysztof; Kendall, Wilfrid S. Efficient Markovian couplings: examples and counterexamples. Ann. Appl. Probab. 10 (2000), no. 2, 362-409. MR1768241 (2002b:60129)

  4. Chen, Xia; Li, Wenbo V. Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion. Ann. Probab. 31 (2003), no. 2, 1052-1077. MR1964958

  5. Corwin, Lawrence J.; Greenleaf, Frederick P. Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge Studies in Advanced Mathematics, 18. Cambridge University Press, Cambridge, 1990. viii+269 pp. ISBN: 0-521-36034-X MR1070979 (92b:22007)

  6. Cranston, M.; Wang, Feng-Yu. A condition for the equivalence of coupling and shift coupling. Ann. Probab. 28 (2000), no. 4, 1666-1679. MR1813838 (2003b:60125)

  7. Temps locaux. (French) [Local times] Exposés du Séminaire J. Azéma-M. Yor. Held at the Université Pierre et Marie Curie, Paris, 1976-1977. With an English summary. Astérisque, 52,53. Société Mathématique de France, Paris, 1978. ii+223 pp. MR0509476 (81b:60042)

  8. Goldstein, Sheldon. Maximal coupling. Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 2, 193-204. MR0516740 (80k:60041)

  9. Griffeath, David. A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 95-106. MR0370771 (51 #6996)

  10. Groeneboom, Piet; Jongbloed, Geurt; Wellner, Jon A. Integrated Brownian motion, conditioned to be positive. Ann. Probab. 27 (1999), no. 3, 1283-1303. MR1733148 (2000i:60092)

  11. Hayes, Thomas P.; Vigoda, Eric. A non-Markovian coloring for randomly sampling colorings. Technical report (2003), Dept. Computer Science, University of Chicago. To appear in FOCS 2003. Math. Review number not available..

  12. Jerrum, Mark. Counting, sampling and integrating: algorithms and complexity. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003. xii+112 pp. ISBN: 3-7643-6946-9 MR1960003 (2004a:68044)

  13. Kaimanovich, V. A. Brownian motion and harmonic functions on covering manifolds. An entropic approach. (Russian) Dokl. Akad. Nauk SSSR 288 (1986), no. 5, 1045-1049. MR0852647 (88k:58163)

  14. Kendall, Wilfrid S. Stochastic differential geometry, a coupling property, and harmonic maps. J. London Math. Soc. (2) 33 (1986), no. 3, 554-566. MR0850971 (88b:58149)

  15. Kendall, Wilfrid S. Probability, convexity, and harmonic maps. II. Smoothness via probabilistic gradient inequalities. J. Funct. Anal. 126 (1994), no. 1, 228-257. MR1305069 (95j:58036)

  16. Khoshnevisan, Davar; Shi, Zhan. Chung's law for integrated Brownian motion. Trans. Amer. Math. Soc. 350 (1998), no. 10, 4253-4264. MR1443196 (98m:60056)

  17. Kolmogoroff, A. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). (German) Ann. of Math. (2) 35 (1934), no. 1, 116-117. MR1503147

  18. Lachal, Aimé. Local asymptotic classes for the successive primitives of Brownian motion. Ann. Probab. 25 (1997), no. 4, 1712-1734. MR1487433 (98j:60112)

  19. Leeb, Bernhard. Harmonic functions along Brownian balls and the Liouville property for solvable Lie groups. Math. Ann. 296 (1993), no. 4, 577-584. MR1233483 (94g:58253)

  20. Lyons, Terry; Sullivan, Dennis. Function theory, random paths and covering spaces. J. Differential Geom. 19 (1984), no. 2, 299-323. MR0755228 (86b:58130)

  21. McKean, H. P., Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 1963 227-235. MR0156389 (27 #6312)

  22. Price, Catherine J. Zeros of Brownian polynomials and Coupling of Brownian areas. Ph. D. thesis (1996), Department of Statistics, University of Warwick. Math. Review number not available.

  23. Propp, James Gary; Wilson, David Bruce. Exact sampling with coupled Markov chains and applications to statistical mechanics. Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995). Random Structures Algorithms 9 (1996), no. 1-2, 223-252. MR1611693 (99k:60176)

  24. Thorisson, Hermann. Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7 MR1741181 (2001b:60003)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.