A New Probability Measure-Valued Stochastic Process with Ferguson-Dirichlet Process as Reversible Measure

Jinghai Shao (Beijing Normal University)


A new diffusion process taking values in the space of all probability measures over $[0,1]$ is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which is the reversible measure of the Fleming-Viot process with parent independent mutation. The intrinsic distance of this process is in the class of Wasserstein distances, so it's also a kind of Wasserstein diffusion. Moreover, this process satisfies the Log-Sobolev inequality.

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Pages: 271-292

Publication Date: January 26, 2011

DOI: 10.1214/EJP.v16-844


  1. Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego. Functions of bounded variation and free discontinuity problems.Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. xviii+434 pp. ISBN: 0-19-850245-1 MR1857292 (2003a:49002)
  2. Chen, Mu-Fa. Eigenvalues, inequalities, and ergodic theory.Probability and its Applications (New York). Springer-Verlag London, Ltd., London, 2005. xiv+228 pp. ISBN: 1-85233-868-7 MR2105651 (2005m:60001)
  3. Döring, Maik; Stannat, Wilhelm. The logarithmic Sobolev inequality for the Wasserstein diffusion. Probab. Theory Related Fields 145 (2009), no. 1-2, 189--209. MR2520126 (2010m:31023)
  4. Ethier, S. N.; Griffiths, R. C. The transition function of a Fleming-Viot process. Ann. Probab. 21 (1993), no. 3, 1571--1590. MR1235429 (95a:60101)
  5. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes.Characterization and convergence.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  6. Ethier, S. N.; Kurtz, Thomas G. Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), no. 2, 345--386. MR1205982 (94d:60131)
  7. Ethier, S. N.; Kurtz, Thomas G. Convergence to Fleming-Viot processes in the weak atomic topology. Stochastic Process. Appl. 54 (1994), no. 1, 1--27. MR1302692 (95m:60075)
  8. Ferguson, Thomas S. A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 (1973), 209--230. MR0350949 (50 #3441)
  9. K. Handa. Quasi-invariance and reversibility in the Fleming-Viot process. Prob. Theory Relat. Fields 122(2002), 545-566. MR1902190 MR1902190
  10. Jordan, Richard; Kinderlehrer, David; Otto, Felix. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998), no. 1, 1--17. MR1617171 (2000b:35258)
  11. Lott, John; Villani, Cédric. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no. 3, 903--991. MR2480619 (2010i:53068)
  12. Ma, Zhi Ming; Röckner, Michael. Introduction to the theory of (nonsymmetric) Dirichlet forms.Universitext. Springer-Verlag, Berlin, 1992. vi+209 pp. ISBN: 3-540-55848-9 MR1214375 (94d:60119)
  13. Otto, F.; Villani, C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), no. 2, 361--400. MR1760620 (2001k:58076)
  14. L. Overbeck, M. Röckner and B. Schmuland. An analytic approach to the Fleiming-Viot processes with interactive selection. Ann. Probab. 23(1995), No. 1, 1-36. MR1330758
  15. A. Schied. Geometric analysis for symmetric Fleming-Viot operators: Rademacher's theorem and exponential families. Potential Analysis 17(2002), 351-374. MR1918241
  16. J.H. Shao. Transportation cost inequalities for Wasserstein diffusions. Preprint.
  17. Shiga, Tokuzo. A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30 (1990), no. 2, 245--279. MR1068791 (91j:60093)
  18. W. Stannat. On the validity of the Log-Sobolev inequality for symmetric Fleming-Viot operators. emph{Ann. Probab.} 28(2000), 667-684. MR1782270
  19. W. Stannat. Functional inequalities for the Wasserstein Dirichlet Form. To appear in Stochastic Analysis, Ascona 2008, Eds.: R. Dalang, et al.
  20. Sturm, Karl-Theodor. On the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65--131. MR2237206 (2007k:53051a)
  21. Sturm, Karl-Theodor. On the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133--177. MR2237207 (2007k:53051b)
  22. 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)
  23. Villani, Cédric. Optimal transport.Old and new.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3 MR2459454 (2010f:49001)
  24. M-K. Von Renesse and K.T. Sturm. Entropy measure and Wasserstein diffusion. Ann. of Probab. 37(2009), 1114-1191. MR2537551
  25. F.-Y. Wang. Functional Inequalities, Markov Semigroups and Spectral Theory. Science Press, Beijing/New York, 2005.

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