Electronic Communications in Probability

A property of Petrov's diffusion

Stewart Ethier

Full-text: Open access


Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor.  We show that the subset of the simplex comprising vectors whose coordinates sum to 1 is the natural state space for the process.  In fact, the complementary set acts like an entrance boundary.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 65, 4 pp.

Accepted: 18 September 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]

infinite-dimensional diffusion process transition density two-parameter Poisson–Dirichlet distribution entrance boundary

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


Ethier, Stewart. A property of Petrov's diffusion. Electron. Commun. Probab. 19 (2014), paper no. 65, 4 pp. doi:10.1214/ECP.v19-3684. https://projecteuclid.org/euclid.ecp/1465316767

Export citation


  • Ethier, S. N. Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Probab. 29 (1992), no. 3, 487–498.
  • Ethier, S. N.; Kurtz, Thomas G. The infinitely-many-neutral-alleles diffusion model. Adv. in Appl. Probab. 13 (1981), no. 3, 429–452.
  • Feng, Shui. The Poisson-Dirichlet distribution and related topics. Models and asymptotic behaviors. Probability and its Applications (New York). Springer, Heidelberg, 2010. xiv+218 pp. ISBN: 978-3-642-11193-8
  • Feng, Shui; Sun, Wei. Some diffusion processes associated with two parameter Poisson-Dirichlet distribution and Dirichlet process. Probab. Theory Related Fields 148 (2010), no. 3-4, 501–525.
  • Feng, Shui; Sun, Wei; Wang, Feng-Yu; Xu, Fang. Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion. J. Funct. Anal. 260 (2011), no. 2, 399–413.
  • Kingman, J. F. C.; Taylor, S. J.; Hawkes, A. G.; Walker, A. M.; Cox, David Roxbee; Smith, A. F. M.; Hill, B. M.; Burville, P. J.; Leonard, T. Random discrete distribution. With a discussion by S. J. Taylor, A. G. Hawkes, A. M. Walker, D. R. Cox, A. F. M. Smith, B. M. Hill, P. J. Burville, T. Leonard and a reply by the author. J. Roy. Statist. Soc. Ser. B 37 (1975), 1–22.
  • Petrov, L. A. A two-parameter family of infinite-dimensional diffusions on the Kingman simplex. (Russian) Funktsional. Anal. i Prilozhen. 43 (2009), no. 4, 45–66; translation in Funct. Anal. Appl. 43 (2009), no. 4, 279–296
  • Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855–900.
  • Schmuland, B. A result on the infinitely many neutral alleles diffusion model. J. Appl. Probab. 28 (1991), no. 2, 253–267.
  • Zhou, Y. Ergodic inequality of three population genetic models. arXiv:1307.0883, 2013.