Journal of Applied Probability

Quasistationary distributions and Fleming-Viot processes in finite spaces

Amine Asselah, Pablo A. Ferrari, and Pablo Groisman

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Abstract

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In the associated Fleming-Viot process N particles evolve independently in Λ with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 / N.

Article information

Source
J. Appl. Probab. Volume 48, Number 2 (2011), 322-332.

Dates
First available in Project Euclid: 21 June 2011

Permanent link to this document
http://projecteuclid.org/euclid.jap/1308662630

Digital Object Identifier
doi:10.1239/jap/1308662630

Zentralblatt MATH identifier
05918586

Mathematical Reviews number (MathSciNet)
MR2840302

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Quasistationary distribution Fleming-Viot process

Citation

Asselah, Amine; Ferrari, Pablo A.; Groisman, Pablo. Quasistationary distributions and Fleming-Viot processes in finite spaces. J. Appl. Probab. 48 (2011), no. 2, 322--332. doi:10.1239/jap/1308662630. http://projecteuclid.org/euclid.jap/1308662630.


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References

  • Bieniek, M., Burdzy, K. and Finch, S. (2009). Non-extinction of a Fleming–Viot particle model. Preprint. Available at http://arxiv.org/abs/0905.1999v1.
  • Burdzy, K., Hołyst, R. and March, P. (2000). A Fleming–Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214, 679–703.
  • Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.
  • Ferrari, P. A. and Marić, N. (2007). Quasi stationary distributions and Fleming–Viot processes in countable spaces. Electron. J. Prob. 12, 684–702.
  • Grigorescu, I. and Kang, M. (2004). Hydrodynamic limit for a Fleming–Viot type system. Stoch. Process. Appl. 110, 111–143.
  • Grigorescu, I. and Kang, M. (2006). Tagged particle limit for a Fleming–Viot type system. Electron. J. Prob. 11, 311–331.
  • Grigorescu, I. and Kang, M. (2011). Immortal particle for a catalytic branching process Prob. Theory Relat. Fields 29pp.
  • Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Prob. 6, 355–378.