Electronic Journal of Probability

Generalised Stable Fleming-Viot Processes as Flickering Random Measures

Matthias Birkner and Jochen Blath

Full-text: Open access

Abstract

We study some remarkable path-properties of generalised stable Fleming-Viot processes (including the so-called spatial Neveu superprocess), inspired by the notion of a "wandering random measure" due to Dawson and Hochberg (1982). In particular, we make use of Donnelly and Kurtz' (1999) modified lookdown construction to analyse their longterm scaling properties, exhibiting a rare natural example of a scaling family of probability laws converging in f.d.d. sense, but not weakly w.r.t. any of Skorohod's topologies on path space. This phenomenon can be explicitly described and intuitively understood in terms of "sparks", leading to the concept of a "flickering random measure". In particular, this completes results of Fleischmann and Wachtel (2006) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 84, 2418-2437.

Dates
Accepted: 3 November 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819545

Digital Object Identifier
doi:10.1214/EJP.v14-712

Mathematical Reviews number (MathSciNet)
MR2563246

Zentralblatt MATH identifier
1190.60040

Subjects
Primary: 60G57: Random measures
Secondary: 60G17: Sample path properties

Keywords
Generalised Fleming-Viot process flickering random measure measure-valued diffusion lookdown construction wandering random measure Neveu superprocess path properties tightness Skorohod topology

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

Citation

Birkner, Matthias; Blath, Jochen. Generalised Stable Fleming-Viot Processes as Flickering Random Measures. Electron. J. Probab. 14 (2009), paper no. 84, 2418--2437. doi:10.1214/EJP.v14-712. https://projecteuclid.org/euclid.ejp/1464819545


Export citation

References

  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
  • Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003), no. 2, 261–288.
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Birkner, Matthias; Blath, Jochen. Measure-valued diffusions, general coalescents and population genetic inference. Trends in Stochastic Analysis, LMS 353, Cambridge University Press, 329–363 (2009).
  • Birkner, Matthias; Blath, Jochen; Capaldo, Marcella; Etheridge, Alison; Möhle, Martin; Schweinsberg, Jason; Wakolbinger, Anton. Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 (2005), no. 9, 303–325 (electronic).
  • Birkner, Matthias; Blath, Jochen; Möhle, Martin; Steinrücken, Matthias; Tams, Johanna. A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 25–61.
  • Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI–-1991, 1–260, Lecture Notes in Math., 1541, Springer, Berlin, 1993.
  • Dawson, Donald A.; Hochberg, Kenneth J. Wandering random measures in the Fleming-Viot model. Ann. Probab. 10 (1982), no. 3, 554–580.
  • Donnelly, Peter; Kurtz, Thomas G. A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24 (1996), no. 2, 698–742.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • El Karoui, Nicole; Roelly, Sylvie. Propriétés de martingales, explosion et représentation de Lévy-Khintchine d'une classe de processus de branchement à valeurs mesures. (French) [Martingale properties, explosion and Levy-Khinchin representation of a class of measure-valued branching processes] Stochastic Process. Appl. 38 (1991), no. 2, 239–266.
  • Etheridge, Alison M. An introduction to superprocesses. University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5
  • Etheridge, Alison; March, Peter. A note on superprocesses. Probab. Theory Related Fields 89 (1991), no. 2, 141–147.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • Feller, William. An introduction to probability theory and its applications. Vol. II. John Wiley & Sons, Inc., New York-London-Sydney 1966 xviii+636 pp.
  • Fleischmann, Klaus; Sturm, Anja. A super-stable motion with infinite mean branching. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 5, 513–537.
  • Fleischmann, Klaus; Wachtel, Vitali. Large scale localization of a spatial version of Neveu's branching process. Stochastic Process. Appl. 116 (2006), no. 7, 983–1011.
  • Fleming, Wendell H.; Viot, Michel. Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 (1979), no. 5, 817–843.
  • Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
  • Möhle, Martin; Sagitov, Serik. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001), no. 4, 1547–1562.
  • Perkins, Edwin A. Conditional Dawson-Watanabe processes and Fleming-Viot processes. Seminar on Stochastic Processes, 1991 (Los Angeles, CA, 1991), 143–156, Progr. Probab., 29, Birkhäuser Boston, Boston, MA, 1992.
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116–1125.
  • Schweinsberg, Jason. A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000), 1–11 (electronic).
  • Skorohod, A. V. Limit theorems for stochastic processes. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 289–319.
  • Yosida, Kôsaku. Functional analysis. Die Grundlehren der Mathematischen Wissenschaften, Band 123 Academic Press, Inc., New York; Springer-Verlag, Berlin 1965 xi+458 pp.