Characterization of maximal Markovian couplings for diffusion processes

Kazumasa Kuwada (Ochanomizu University)

Abstract


Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufficient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we find many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we find a Markov chain which admits no maximal Markovian coupling for specified starting points.

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

Pages: 633-662

Publication Date: March 10, 2009

DOI: 10.1214/EJP.v14-634

References

  1. Barlow, Martin T.; Bass, Richard F.; Kumagai, Takashi. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58 (2006), no. 2, 485--519. MR2228569 (2007f:60064)
  2. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6 MR1835418 (2002e:53053)
  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. Chavel, Isaac. Riemannian geometry---a modern introduction. Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge, 1993. xii+386 pp. ISBN: 0-521-43201-4; 0-521-48578-9 MR1271141 (95j:53001)
  5. Chen, Mu Fa; Wang, Feng Yu. Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37 (1994), no. 1, 1--14. MR1308707 (96d:58141)
  6. Cranston, M. Gradient estimates on manifolds using coupling. J. Funct. Anal. 99 (1991), no. 1, 110--124. MR1120916 (93a:58175)
  7. Goldstein, Sheldon. Maximal coupling. Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 2, 193--204. MR0516740 (80k:60041)
  8. Griffeath, David. A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 95--106. MR0370771 (51 #6996)
  9. Iwahori, Nagayoshi. On discrete reflection groups on symmetric Riemannian manifolds. 1966 Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) pp. 57--62 Nippon Hyoronsha, Tokyo MR0217741 (36 #830)
  10. Kendall, Wilfrid S. Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19 (1986), no. 1-2, 111--129. MR0864339 (88e:60092)
  11. Kobayashi, Shoshichi; Nomizu, Katsumi. Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp. MR0238225 (38 #6501)
  12. Kumagai, Takashi. Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals. Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 223--240. MR1442498 (98k:60130)
  13. Kuwada, Kazumasa. On uniqueness of maximal coupling for diffusion processes with a reflection. J. Theoret. Probab. 20 (2007), no. 4, 935--957. MR2359063 (2008i:60132)
  14. Kuwae, Kazuhiro; Machigashira, Yoshiroh; Shioya, Takashi. Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. Math. Z. 238 (2001), no. 2, 269--316. MR1865418 (2002m:58052)
  15. Kuwae, Kazuhiro; Shioya, Takashi. Laplacian comparison for Alexandrov spaces. preprint
  16. Lindvall, Torgny. Lectures on the coupling method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1992. xiv+257 pp. ISBN: 0-471-54025-0 MR1180522 (94c:60002)
  17. Norris, James R. Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997), no. 1, 79--103. MR1484769 (99d:58167)
  18. Pitman, J. W. On coupling of Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 4, 315--322. MR0415775 (54 #3854)
  19. Saloff-Coste, Laurent. Aspects of Sobolev-type inequalities. London Mathematical Society Lecture Note Series, 289. Cambridge University Press, Cambridge, 2002. x+190 pp. ISBN: 0-521-00607-4 MR1872526 (2003c:46048)
  20. Spitzer, Frank. Principles of random walks. Second edition. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp. MR0388547 (52 #9383)
  21. Sturm, Karl-Theodor. Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995), no. 2, 275--312. MR1355744 (97b:35003)
  22. Sturm, K. T. Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 (1996), no. 3, 273--297. MR1387522 (97k:31010)
  23. Sverchkov, M. Yu.; Smirnov, S. N. Maximal coupling for processes in $D[0,\infty]$. (Russian) Dokl. Akad. Nauk SSSR 311 (1990), no. 5, 1059--1061; translation in Soviet Math. Dokl. 41 (1990), no. 2, 352--354 (1991) MR1072656 (91i:60099)
  24. Varopoulos, N. Th. Small time Gaussian estimates of heat diffusion kernels. II. The theory of large deviations. J. Funct. Anal. 93 (1990), no. 1, 1--33. MR1070036 (91j:58156)
  25. Villani, Cédric. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. xvi+370 pp. ISBN: 0-8218-3312-X MR1964483 (2004e:90003)
  26. von Renesse, Max-K. Heat kernel comparison on Alexandrov spaces with curvature bounded below. Potential Anal. 21 (2004), no. 2, 151--176. MR2058031 (2005m:58055)
  27. von Renesse, Max-K. Intrinsic coupling on Riemannian manifolds and polyhedra. Electron. J. Probab. 9 (2004), no. 14, 411--435 (electronic). MR2080605 (2005i:60158)
Bookmark and Share


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