Electronic Journal of Probability

Tagged Particle Limit for a Fleming-Viot Type System

Ilie Grigorescu and Min Kang

Full-text: Open access

Abstract

We consider a branching system of $N$ Brownian particles evolving independently in a domain $D$ during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles, the complement of the set $D$ acting as a catalyst or hard obstacle. Identifying the newly born particle with the one killed upon contact with the catalyst, we determine the exact law of the tagged particle as $N$ approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit. Both results can be seen as scaling limits of a genome population undergoing redistribution present in the Fleming-Viot dynamics.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 12, 311-331.

Dates
Accepted: 20 April 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730547

Digital Object Identifier
doi:10.1214/EJP.v11-316

Mathematical Reviews number (MathSciNet)
MR2217819

Zentralblatt MATH identifier
1109.60083

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J50: Boundary theory 35K15: Initial value problems for second-order parabolic equations

Keywords
Fleming-Viot propagation of chaos tagged particle

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Grigorescu, Ilie; Kang, Min. Tagged Particle Limit for a Fleming-Viot Type System. Electron. J. Probab. 11 (2006), paper no. 12, 311--331. doi:10.1214/EJP.v11-316. https://projecteuclid.org/euclid.ejp/1464730547


Export citation

References

  • Billingsley, P. Convergence of Probability Measures. Wiley series in probability and statistics, New York (1968)
  • Burdzy, K., Hol yst, R., Ingerman, D., March, P. (1996) Configurational transition in a Fleming-Viot type model and probabilistic interpretation of Laplacian eigenfunctions J. Phys. A 29, 2633-2642.
  • Burdzy, K., Hol yst, R., March, P. (2000) A Fleming-Viot particle representation of the Dirichlet Laplacian Comm. Math. Phys. 214, no. 3.
  • Dawson, D.A. (1992) Infinitely divisible random measures and superprocesses. In: Stochastic Analysis and Related Topics, H. Körezlioglu and A.S. Üstünel, Eds, Boston: Birkh"a"user.
  • Ethier, S., Kurtz, T. (1986) Markov processes : characterization and convergence. Wiley series in probability and statistics, New York.
  • Evans, L.C. (1998) Partial Differential Equations American Mathematical Society, Providence, R.I.
  • Grigorescu, I., Kang, M. (2002) Brownian motion on the figure eight Journal of Theoretical Probability, 15 (3): 817-844.
  • Grigorescu, I., Kang, M. (2003) Path Collapse for an Inhomogeneous Random Walk. J. Theoret. Probab. 16, no. 1, 147–159.
  • Grigorescu, I., Kang, M. (2005) Ergodic Properties of Multidimensional Brownian Motion with Rebirth Preprint. Preprint.
  • Grigorescu, Ilie; Kang, Min Path collapse for multidimensional Brownian motion with rebirth. Statist. Probab. Lett. 70 (2004), no. 3, 199–209.
  • Grigorescu, I., Kang, M. (2004) Hydrodynamic Limit for a Fleming-Viot Type System. Stochastic Process. Appl. 110, no. 1, 111-143.
  • Hiraba, S.(2000) Jump-type Fleming-Viot processes }Adv. in Appl. Probab. 32, no. 1, 140–158.
  • Ikeda, N., Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes Second Edition, North-Holland, Amsterdam and Kodansha, Tokyo.
  • Oelschl"a"ger, K. (1985) A law of large numbers for moderately interacting diffusion processes Z. Wahrscheinlichkeitstheorie verw. Gebiete, vol 69, 279-322.
  • Kipnis, C.; Landim, C. (1999) Scaling Limits of Interacting Particle Systems} Springer-Verlag, New York.