Poisson representations of branching Markov and measure-valued branching processes. (English) Zbl 1232.60053

The authors give some representations of branching Markov processes and their measure-valued limits in terms of countable particle systems. A particle in a system has a location and a level. The results improve ealier ones of the first author and his collaborators. In particular, for models with branching rates independent of the particle location, a similar approach was carried out in the earlier work [P. Donnelly and T. G. Kurtz, “Particle representations for measure-valued population models”, Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)]. The result was extended to general branching rates and critical or subcritical offspring distributions in [T. G. Kurtz, “Particle representations for measure-valued population processes with spatially varying birth rates”, CMS Conf. Proc. 26, 299–317 (2000; Zbl 0960.60069)]. The representation given in the present paper applies to both subcritical and supercritical processes as well as to processes with heavy-tailed offspring distributions. It also applies to models with infinite mass. The levels of the particles in the model considered here change with time. In fact, before taking the limit, death of a particle occurs only when the level of the particle crosses a specified level \(r\geq 0\). For the limiting models, death occurs only when the level goes to infinity. As in the earlier work, the justification for the representation is a consequence of a Markov mapping theorem. For branching Markov processes, at each time \(t\geq 0\), conditioned on the state of the process, the levels are independent and uniformly distributed on \([0,r]\). For the limiting measure-valued process, at each time \(t\geq 0\), the joint distribution of locations and levels is conditionally Poisson distributed with mean measure \(K(t)\times \Lambda\), where \(\Lambda\) denotes the Lebesgue measure, and \(K(t)\) is the desired measure-valued process. The representation simplifies or gives new proofs for a variety of results including conditioning on extinction or nonextinction, Harris’s convergence theorem and diffusion approximations. In particular, the Feller diffusion approximation for nearly critical branching processes is obtained as a consequence of the construction.


60J68 Superprocesses
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J60 Diffusion processes
60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI arXiv


[1] Berestycki, J., Berestycki, N. and Limic, V. The \lambda -coalescent speed of coming down from infinity. · Zbl 1247.60110
[2] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147-181 (electronic). · Zbl 1110.60026
[3] Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete 60 185-201. · Zbl 0468.60034
[4] Dawson, D. A. (1975). Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5 1-52. · Zbl 0299.60050
[5] Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24 698-742. · Zbl 0869.60074
[6] Donnelly, P. and Kurtz, T. G. (1999). Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9 1091-1148. · Zbl 0964.60075
[7] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166-205. · Zbl 0956.60081
[8] Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a measure-valued branching diffusion with immigration. In Stochastic Processes 71-79. Springer, New York. · Zbl 0783.60077
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[10] Evans, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959-971. · Zbl 0784.60052
[11] Evans, S. N. and O’Connell, N. (1994). Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. 37 187-196. · Zbl 0804.60042
[12] Evans, S. N. and Perkins, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 329-337. · Zbl 0717.60099
[13] Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 337-361 (1989). · Zbl 0673.60089
[14] Gorostiza, L. G. and López-Mimbela, J. A. (1990). The multitype measure branching process. Adv. in Appl. Probab. 22 49-67. · Zbl 0711.60084
[15] Grey, D. R. (1988). Supercritical branching processes with density independent catastrophes. Math. Proc. Cambridge Philos. Soc. 104 413-416. · Zbl 0657.60108
[16] Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Probab. 2 1027-1045. · Zbl 0361.60062
[17] Harris, T. E. (1951). Some mathematical models for branching processes. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability , 1950 305-328. Univ. of California Press, Berkeley and Los Angeles. · Zbl 0045.07701
[18] Helland, I. S. (1981). Minimal conditions for weak convergence to a diffusion process on the line. Ann. Probab. 9 429-452. · Zbl 0459.60027
[19] Hering, H. (1978). The non-degenerate limit for supercritical branching diffusions. Duke Math. J. 45 561-600. · Zbl 0391.60078
[20] Hering, H. and Hoppe, F. M. (1981). Critical branching diffusions: Proper normalization and conditioned limit. Ann. Inst. H. Poincaré Sect. B ( N.S. ) 17 251-274. · Zbl 0467.60075
[21] Ikeda, N., Nagasawa, M. and Watanabe, S. (1965). On branching Markov processes. Proc. Japan Acad. 41 816-821. · Zbl 0224.60038
[22] Ikeda, N., Nagasawa, M. and Watanabe, S. (1968). Branching Markov processes. I. J. Math. Kyoto Univ. 8 233-278. · Zbl 0233.60068
[23] Ikeda, N., Nagasawa, M. and Watanabe, S. (1968). Branching Markov processes. II. J. Math. Kyoto Univ. 8 365-410. · Zbl 0233.60069
[24] Ikeda, N., Nagasawa, M. and Watanabe, S. (1969). Branching Markov processes. III. J. Math. Kyoto Univ. 9 95-160. · Zbl 0233.60070
[25] Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 20-65. JSTOR: · Zbl 0595.60008
[26] Keiding, N. (1975). Extinction and exponential growth in random environments. Theoret. Population Biol. 8 49-63. · Zbl 0311.92019
[27] Kulperger, R. (1979). Brillinger type mixing conditions for a simple branching diffusion process. Stochastic Process. Appl. 9 55-66. · Zbl 0411.60087
[28] Kurtz, T. G. (1973). A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12 55-67. · Zbl 0246.47053
[29] Kurtz, T. G. (1978). Diffusion approximations for branching processes. In Branching Processes ( Conf. , Saint Hippolyte , Que. , 1976). Adv. Probab. Related Topics 5 269-292. Dekker, New York. · Zbl 0405.60080
[30] Kurtz, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3 29 pp. (electronic). · Zbl 0907.60065
[31] Kurtz, T. G. (2000). Particle representations for measure-valued population processes with spatially varying birth rates. In Stochastic Models ( Ottawa , ON , 1998). CMS Conf. Proc. 26 299-317. Amer. Math. Soc., Providence, RI. · Zbl 0960.60069
[32] Kurtz, T. G. and Nappo, G. (2010). The filtered martingale problem. In Handbook on Nonlinear Filtering (D. Crisan and B. Rozovsky, eds.). Oxford Univ. Press. · Zbl 1229.60048
[33] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053
[34] Kurtz, T. G. and Stockbridge, R. H. (2001). Stationary solutions and forward equations for controlled and singular martingale problems. Electron. J. Probab. 6 52 pp. (electronic). · Zbl 0984.60086
[35] Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103-126. · Zbl 0996.60071
[36] Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Teor. Verojatnost. i Primenen. 13 126-137. · Zbl 0253.60073
[37] Li, Z. H. (1992). Measure-valued branching processes with immigration. Stochastic Process. Appl. 43 249-264. · Zbl 0760.60075
[38] Li, Z. H., Li, Z. B. and Wang, Z. K. (1993). Asymptotic behavior of the measure-valued branching process with immigration. Sci. China Ser. A 36 769-777. · Zbl 0790.60068
[39] Li, Z. and Wang, Z. (1999). Measure-valued branching processes and immigration processes. Adv. Math. ( China ) 28 105-134. · Zbl 1054.60517
[40] Mellein, B. (1982). Diffusion limits of conditioned critical Galton-Watson processes. Rev. Colombiana Mat. 16 125-140. · Zbl 0506.60086
[41] Pakes, A. G. (1986). The Markov branching-catastrophe process. Stochastic Process. Appl. 23 1-33. · Zbl 0633.92014
[42] Pakes, A. G. (1987). Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biol. 25 307-325. · Zbl 0642.92012
[43] Pakes, A. G. (1988). The Markov branching process with density-independent catastrophes. I. Behaviour of extinction probabilities. Math. Proc. Cambridge Philos. Soc. 103 351-366. · Zbl 0641.60089
[44] Pakes, A. G. (1989). Asymptotic results for the extinction time of Markov branching processes allowing emigration. I. Random walk decrements. Adv. in Appl. Probab. 21 243-269. JSTOR: · Zbl 0671.60078
[45] Pakes, A. G. (1989). The Markov branching process with density-independent catastrophes. II. The subcritical and critical cases. Math. Proc. Cambridge Philos. Soc. 106 369-383. · Zbl 0735.60090
[46] Pakes, A. G. (1990). The Markov branching process with density-independent catastrophes. III. The supercritical case. Math. Proc. Cambridge Philos. Soc. 107 177-192. · Zbl 0768.60073
[47] Schweinsberg, J. (2000). A necessary and sufficient condition for the \Lambda -coalescent to come down from infinity. Electron. Comm. Probab. 5 1-11 (electronic). · Zbl 0953.60072
[48] Stannat, W. (2003). On transition semigroups of ( A ,\Psi )-superprocesses with immigration. Ann. Probab. 31 1377-1412. · Zbl 1043.60076
[49] Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 141-167. · Zbl 0159.46201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.