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Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Amine Asselah ,
Pablo A. Ferrari  and
Pablo Groisman
Amine Asselah*
Affiliation:
Université Paris-Est
Pablo A. Ferrari*
Affiliation:
Universidade de São Paulo and Universidad de Buenos Aires
Pablo Groisman*
Affiliation:
Universidad de Buenos Aires
*
∗Postal address: LAMA, Université Paris-Est, CNRS UMR 8050, 61 Avenue General de Gaulle, 94010 Creteil Cedex, France.
∗∗Postal address: DM-FCEN, Universidad de Buenos Aires, Pabellon 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina.
∗∗Postal address: DM-FCEN, Universidad de Buenos Aires, Pabellon 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina.
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Abstract

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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.

Keywords

Quasistationary distributionFleming-Viot process

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 322 - 332
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] 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.Google Scholar
[2] Burdzy, K., Hołyst, R. and March, P. (2000). A Fleming–Viot particle representation of the Dirichlet Laplacian. Commun. Math. Phys. 214, 679–703.Google Scholar
[3] Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle Scholar
[4] Ferrari, P. A. and Marić, N. (2007). Quasi stationary distributions and Fleming–Viot processes in countable spaces. Electron. J. Prob. 12, 684–702.CrossRefGoogle Scholar
[5] Grigorescu, I. and Kang, M. (2004). Hydrodynamic limit for a Fleming–Viot type system. Stoch. Process. Appl. 110, 111–143.CrossRefGoogle Scholar
[6] Grigorescu, I. and Kang, M. (2006). Tagged particle limit for a Fleming–Viot type system. Electron. J. Prob. 11, 311–331.CrossRefGoogle Scholar
[7] Grigorescu, I. and Kang, M. (2011). Immortal particle for a catalytic branching process Prob. Theory Relat. Fields 29pp.Google Scholar
[8] Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Prob. 6, 355–378.CrossRefGoogle Scholar