Abstract.
We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.
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Blath, J., Etheridge, A.M., Meredith, M.E.: Coexistence in locally regulated competing populations. University of Oxford, Preprint, 2003
Barton, N.H., Etheridge, A.M., Sturm, A.K.: Coalescence in a random background. WIAS Berlin, Preprint No. 756, 2002. To appear in Ann. Appl. Probab.
Chen, M.F.: Existence theorems for interacting particle systems with non-compact state space. Sci. China Ser. A 30, 148–156 (1987)
Dawson, D.A.: Measure-valued Markov processes. In: P.L. Hennequin (ed.), École d’été de probabilités de Saint Flour XXI–1991, volume 1541 of Lecture Notes in Mathematics, Springer, Berlin, 1993, pp. 1–260
Dawson, D.A., Greven, A.: Hierarchically interacting Fleming-Viot processes with selection and mutation: Multiple space time scale analysis and quasi-equilibria. Electron. J. Probab.4 (4), 81 (1999)
Ding, W., Durrett, R., Liggett, T.M.: Ergodicity of reversible reaction diffusion processes. Probab. Theory Relat. Fields 85 (1), 13–26 (1990)
Donelly, P., Kurtz, T.G.: Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9, 1091–1148 (1999)
Ethier, S.N., Kurtz, T.G.: Markov Processes; Characterization and Convergence. John Wiley & Sons, New York, 1986
Fleischmann, K., Swart, J.M. Extinction versus exponential growth in a supercritical super-Wright-Fisher diffusion. Stochastic Processes Appl. 106 (1), 141–165 (2003)
Fleischmann, K., Swart, J.M. Trimmed trees and embedded particle systems. WIAS Berlin, Preprint No. 793, 2002. To appear in Ann. Probab.
Kallenberg, O.: Random measures. 3rd rev. and enl. ed. Akademie-Verlag, Berlin, 1983
Krone, S.M., Neuhauser, C.: Ancestral processes with selection. Theor. Popul. Biol. 51 (3), 210–237 (1997)
Liggett, T.M.: Interacting Particle Systems. Springer-Verlag, New York, 1985
Liggett, T.M. Stochastic Interacting Systems: Contact, Voter and Exclusion Process. Springer-Verlag, Berlin, 1999
Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. verw. Gebiete 56, 443–468 (1981)
Mountford, T.S.: The ergodicity of a class of reversible reaction-diffusion processes. Probab. Theory Relat. Fields 92 (2), 259–274 (1992)
Müller, C., Tribe, R.: Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Relat. Fields 102 (4), 519–545 (1995)
Neuhauser, C.: An ergodic theorem for Schlögl models with small migration. Probab. Theory Relat. Fields 85 (1), 27–32 (1990)
Schlögl, F.: Chemical reaction models and non-equilibrium phase transitions. Z. Phys. 253, 147–161 (1972)
Shiga, T.: Diffusion processes in population genetics. J. Math. Kyoto Univ. 21, 133–151 (1981)
Shiga, T., Shimizu, A.: Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20, 395–416 (1980)
Shiga, T., Uchiyama, K.: Stationary states and their stability of the stepping stone model involving mutation and selection. Probab. Theory Relat. Fields 73, 87–117 (1986)
Swart, J.M.: Large Space-Time Scale Behavior of Linearly Interacting Diffusions. PhD thesis, Katholieke Universiteit Nijmegen, 1999 http://helikon.ubn.kun.nl/ mono/s/swart_j/largspscb.pdf.
Swart, J.M.: Clustering of linearly interacting diffusions and universality of their long-time limit distribution. Probab. Theory Relat. Fields 118, 574–594 (2000)
Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167 (1971)
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Mathematics Subject Classification (2000):Primary: 60K35, 92D25; Secondary: 60J80, 60J60
Research supported in part by the German Science Foundation.
AcknowledgementWe thank Klaus Fleischmann who played a stimulating role during the early stages of this project and answered a question about Laplace functionals, Claudia Neuhauser for answering questions about branching-coalescing processes, Olle Häggström for answering questions on nonamenable groups, and Tokuzo Shiga for answering our questions about his work. We thank the referee for drawing our attention to the reference [SU86]. Part of this work was carried out during the visits of Siva Athreya to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin and to the Friedrich-Alexander University Erlangen-Nuremberg, and of Jan Swart to the Indian Statistical Institute, Delhi. We thank all these places for their kind hospitality.
An erratum to this article is available at http://dx.doi.org/10.1007/s00440-009-0232-8.
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Arthreya, S., Swart, J. Branching-coalescing particle systems. Probab. Theory Relat. Fields 131, 376–414 (2005). https://doi.org/10.1007/s00440-004-0377-4
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DOI: https://doi.org/10.1007/s00440-004-0377-4