Construction of markovian coalescents
Annales de l'I.H.P. Probabilités et statistiques, Volume 34 (1998) no. 3, p. 339-383
@article{AIHPB_1998__34_3_339_0,
     author = {Evans, Steven N. and Pitman, Jim},
     title = {Construction of markovian coalescents},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {34},
     number = {3},
     year = {1998},
     pages = {339-383},
     zbl = {0906.60058},
     mrnumber = {1625867},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_1998__34_3_339_0}
}
Evans, Steven N.; Pitman, Jim. Construction of markovian coalescents. Annales de l'I.H.P. Probabilités et statistiques, Volume 34 (1998) no. 3, pp. 339-383. http://www.numdam.org/item/AIHPB_1998__34_3_339_0/

[1] D. Aldous. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab., Vol. 25, 1997, pp. 812-854. | MR 1434128 | Zbl 0877.60010

[2] D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, Vol. 5, 1994, pp. 487-512. | MR 1293075 | Zbl 0811.60057

[3] D. Aldous and J. Pitman. The entrance boundary of the additive coalescent. Paper in preparation, 1997.

[4] D.J. Aldous. Exchangeability and related topics. In P.L. Hennequin, editor, École d'Été de Probabilités de Saint-Flour XII, Springer Lecture Notes in Mathematics, Vol. 1117. Springer-Verlag, 1985. | MR 883646 | Zbl 0562.60042

[5] D.J. Aldous. The continuum random tree III. Ann. Probab., Vol. 21, 1993, pp. 248-289. | MR 1207226 | Zbl 0791.60009

[6] D.J. Aldous. Deterministic and stochastic models for coalescence: a review of the mean-field theory for probabilists. To appear in Bernoulli. Available via homepage http://www.stat.berkeley.edu/users/aldous, 1997. | MR 1673235 | Zbl 0930.60096

[7] D.J. Aldous and V. Limic. The entrance boundary of the multiplicative coalescent. Unpublished. Available via homepage http://www.stat.berkeley.edu/users/aldous, 1997. | Zbl 0889.60080

[8] D.J. Aldous and J. Pitman. The standard additive coalescent. Technical Report 489, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman.

[9] A. Benveniste and J. Jacod. Systèmes de Levy des processus de Markov. Invent. Math., Vol. 21, 1973, pp. 183-198. | MR 343375 | Zbl 0265.60074

[10] R.M. Blumenthal and R.K. Getoor. Markov Processes and Potential Theory. Academic Press, New York, 1968. | MR 264757 | Zbl 0169.49204

[11] E. Bolthausen and A.-S. Sznitman. On Ruelle's probability cascades and an abstract cavity method. To appear in Comm. Math. Phys., 1998. | MR 1652734 | Zbl 0927.60071

[12] A. Broder. Generating random spanning trees. In Proc. 30'th IEEE Symp. Found. Comp. Sci., 1989, pp. 442-447.

[13] A. Cayley. A theorem on trees. Quarterly Journal of Pure and Applied Mathematics, 23:376-378, 1889. (Also in The Collected Mathematical Papers of Arthur CAYLEY. Vol. XIII, pp. 26-28, Cambridge University Press, 1897). | JFM 21.0687.01

[14] D.A. Dawson. Measure-valued Markov processes. In École d'Été de Probabilités de Saint-Flour, 1991, Springer Lecture Notes in Mathematics, Vol. 1117, 1994. | MR 1242575 | Zbl 0799.60080

[15] P. Donnelly and P. Joyce. Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stochastic Processes and their Applications, Vol. 31, 1989, pp. 89-103. | MR 996613 | Zbl 0694.60009

[16] P. Donnelly and S. Simons. On the stochastic approach to cluster size distribution during particle coagulation I: Asymptotic expansion in the deterministic limit. J. Phys. A: Math. Gen., Vol. 26, 1993, pp. 2755-2767. | MR 1236145 | Zbl 0788.60127

[17] S.N. Ethier and T.G. Kurtz. Fleming-Viot processes in population genetics. SIAM Journal on Control and Optimization, Vol. 31, 1993, pp. 345-386. | MR 1205982 | Zbl 0774.60045

[18] S.N. Evans. Infinitely-many-species Lotka-Volterra equations arising from systems of coalescing masses. Technical Report 478, Dept. Statistics, U.C. Berkeley, 1997. To appear in J. London Math. Soc.. Available via http://www.stat.berkeley.edu/tech-reports. | MR 1721823 | Zbl 0940.35051

[19] S.N. Evans and J. Pitman. Stationary Markov processes related to stable Ornstein-Uhlenbeck processes and the additive coalescent. Technical Report 488, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman. | MR 1649003

[20] E.M. Hendriks, J.L. Spouge, M. Eibl, and M. Shreckenberg. Exact solutions for random coagulation processes. Z. Phys. B - Condensed Matter, Vol. 58, 1985, pp. 219-227.

[21] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North Holland/Kodansha, 1989. 2nd edition. | MR 1011252 | Zbl 0684.60040

[22] J.F.C. Kingman. Random discrete distributions. J. Roy. Statist. Soc. B, Vol. 37, 1975, pp. 1-22 | MR 368264 | Zbl 0331.62019

[23] J.F.C. Kingman. The representation of partition structures. J. London Math. Soc., Vol. 18, 1978, pp. 374-380. | MR 509954 | Zbl 0415.92009

[24] J.F.C. Kingman. The coalescent. Stochastic Processes and their Applications, Vol. 13, 1982, pp. 235-248. | MR 671034 | Zbl 0491.60076

[25] J.F.C. Kingman. On the genealogy of large populations. In J. Gani and E.J. Hannan, editors, Essays in Statistical Science, volume 19A of J.A.P. Special vol., pp. 27-43. Applied Probability Trust, 1982. | MR 633178 | Zbl 0516.92011

[26] C. Lacey and S. Cole. Merger rates in hierarchical models of galaxy formation. Mon. Not. R. Astron. Soc., Vol. 262, 1993, pp. 627-649.

[27] A.A. Lushnikov. Coagulation in finite systems. J. Colloid and Interface Science, Vol. 65, 1978, 276-285.

[28] R. Lyons and Y. Peres. Probability on trees and networks. Book in preparation, available at http://www.ma.huji.ac.il/ lyons/prbtree.html, 1996.

[29] A.H. Marcus. Stochastic coalescence. Technometrics, Vol. 10, 1968, pp. 133-143. | MR 223151

[30] M. Perman. Order statistics for jumps of normalized subordinators. Stoch. Proc. Appl., Vol. 46, 1993, pp. 267-281. | MR 1226412 | Zbl 0777.60070

[31] M. Perman, J. Pitman, and M. Yor. Size-biased sampling of Poisson point processes and excursions. Probab. Th. Rel. Fields, Vol. 92, 1992, pp. 21-39. | MR 1156448 | Zbl 0741.60037

[32] J. Pitman. Exchangeable and partially exchangeable random partitions. Probab. Th. Rel. Fields, Vol. 102, 1995, pp. 145-158. | MR 1337249 | Zbl 0821.60047

[33] J. Pitman. Coalescent random forests. Technical Report 457, Dept. Statistics, U.C. Berkeley, 1996. Available via http://www.stat.berkeley.edu/users/pitman.

[34] J. Pitman. Random discrete distributions invariant under size-biased permutation. Adv. Appl. Prob., Vol. 28, 1996, pp. 525-539. | MR 1387889 | Zbl 0853.62018

[35] J. Pitman. Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests and subsets. Technical Report 498, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman.

[36] J. Pitman. Coalescents with multiple collisions. Technical Report 495, Dept. Statistics, U.C. Berkeley, 1997. Available via http://www.stat.berkeley.edu/users/pitman.

[37] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., Vol. 25, 1997, pp. 855-900. | MR 1434129 | Zbl 0880.60076

[38] A. Rényi. On the enumeration of trees. In R. Guy, H. Hanani, N. Sauer, and J. Schonheim, editors, Combinatorial Structures and their Applications, pages 355-360. Gordon and Breach, New York, 1970. | MR 263708 | Zbl 0247.05102

[39] R.K. Sheth. Merging and hierarchical clustering from an initially Poisson distribution. Mon. Not. R. Astron. Soc., Vol. 276, 1995, pp. 796-824.

[40] R.K. Sheth and J. Pitman. Coagulation and branching process models of gravitational clustering. Mon. Not. R. Astron. Soc., Vol. 289, 1997, pp. 66-80.

[41] R. Stanley. Enumerative combinatorics, vol. 2. Book in preparation, to be published by Cambridge University Press, 1996. | MR 1676282

[42] H. Tanaka and K. Nakazawa. Stochastic coagulation equation and validity of the statistical coagulation equation. J. Geomag. Geoelectr., Vol. 45, 1993, pp. 361-381.

[43] H. Tanaka and K. Nakazawa. Validity of the stochastic coagulation equation and runaway growth of protoplanets. Icarus, Vol. 107, 1994, pp. 404-412.

[44] H.M. Taylor and S. Karlin.An Introduction to Stochastic Modeling (revised ed.). Academic Press, Boston, 1994. | MR 1242559 | Zbl 0796.60001

[45] P.G.J. Van Dongen. Fluctuations in coagulating systems II. J. Statist. Phys., Vol. 49, 1987, pp. 927-975. | MR 935481

[46] P.G.J. Van Dongen and M.H. Ernst. Fluctuations in coagulating systems. J. Statist. Phys., Vol. 49, 1987, pp. 879-926. | MR 935480

[47] M. Von Smoluchowski. Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Physik. Z., Vol. 17, 1916, pp. 557-585.