Extinction of Fleming-Viot-type particle systems with strong drift

Mariusz Bieniek (Maria Curie-Skƚodowska University)
Krzysztof Burdzy (University of Washington)
Soumik Pal (University of Washington)


We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. If there are only two particles and the underlying motion is a Bessel process on $(0,\infty)$, both particles converge to 0 at a finite time if and only if the dimension of the Bessel process is less than 0. If the underlying diffusion is Brownian motion with a drift stronger than (but arbitrarily close to, in a suitable sense) the drift of a Bessel process, all particles converge to 0 at a finite time, for any number of particles.

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

Pages: 1-15

Publication Date: January 29, 2012

DOI: 10.1214/EJP.v17-1770


  • Handbook of mathematical functions with formulas, graphs, and mathematical tables. Edited by Milton Abramowitz and Irene A. Stegun. Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. xiv+1046 pp. ISBN: 0-486-61272-4 MR1225604
  • Bass, Richard F. Diffusions and elliptic operators. Probability and its Applications (New York). Springer-Verlag, New York, 1998. xiv+232 pp. ISBN: 0-387-98315-5 MR1483890
  • M. Bieniek, K. Burdzy, and S. Finch. Non-extinction of a Fleming-Viot particle model. Probability Theory and Related Fields, pages 1-40. 10.1007/s00440-011-0372-5.
  • Bougerol, Philippe; Picard, Nico. Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), no. 4, 1714--1730. MR1188039
  • Burdzy, Krzysztof; Hołyst, Robert; March, Peter. A Fleming-Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys. 214 (2000), no. 3, 679--703. MR1800866
  • Diaconis, Persi; Freedman, David. Iterated random functions. SIAM Rev. 41 (1999), no. 1, 45--76. MR1669737
  • Freidlin, M. I.; Wentzell, A. D. Random perturbations of dynamical systems. Translated from the 1979 Russian original by Joseph Szücs. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer-Verlag, New York, 1998. xii+430 pp. ISBN: 0-387-98362-7 MR1652127
  • Göing-Jaeschke, Anja; Yor, Marc. A survey and some generalizations of Bessel processes. Bernoulli 9 (2003), no. 2, 313--349. MR1997032
  • Goldie, Charles M.; Maller, Ross A. Stability of perpetuities. Ann. Probab. 28 (2000), no. 3, 1195--1218. MR1797309
  • I. Grigorescu and M. Kang. Immortal particle for a catalytic branching process. Probability Theory and Related Fields, pages 1-29. 10.1007/s00440-011-0347-6.
  • S. Pal. Wright-Fisher model with negative mutation rates. Ann. Probab., 2011. To appear.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Shiga, Tokuzo; Watanabe, Shinzo. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 37--46. MR0368192
  • Stein, Elias M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp. MR0290095

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