The Annals of Probability

Poisson representations of branching Markov and measure-valued branching processes

Thomas G. Kurtz and Eliane R. Rodrigues

Full-text: Open access


Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov processes, at each time t, 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, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure K(tΛ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process.

The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

Article information

Ann. Probab., Volume 39, Number 3 (2011), 939-984.

First available in Project Euclid: 16 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments

Branching Markov process Dawson–Watanabe process superprocess measure-valued diffusion particle representation Feller diffusion exchangeability Cox process conditioning random environments


Kurtz, Thomas G.; Rodrigues, Eliane R. Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 (2011), no. 3, 939--984. doi:10.1214/10-AOP574.

Export citation


  • [1] Berestycki, J., Berestycki, N. and Limic, V. The λ-coalescent speed of coming down from infinity. Preprint.
  • [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).
  • [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.
  • [4] Dawson, D. A. (1975). Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5 1–52.
  • [5] Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24 698–742.
  • [6] Donnelly, P. and Kurtz, T. G. (1999). Genealogical processes for Fleming–Viot models with selection and recombination. Ann. Appl. Probab. 9 1091–1148.
  • [7] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [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.
  • [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [10] Evans, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959–971.
  • [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.
  • [12] Evans, S. N. and Perkins, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 329–337.
  • [13] Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 337–361 (1989).
  • [14] Gorostiza, L. G. and López-Mimbela, J. A. (1990). The multitype measure branching process. Adv. in Appl. Probab. 22 49–67.
  • [15] Grey, D. R. (1988). Supercritical branching processes with density independent catastrophes. Math. Proc. Cambridge Philos. Soc. 104 413–416.
  • [16] Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Probab. 2 1027–1045.
  • [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.
  • [18] Helland, I. S. (1981). Minimal conditions for weak convergence to a diffusion process on the line. Ann. Probab. 9 429–452.
  • [19] Hering, H. (1978). The non-degenerate limit for supercritical branching diffusions. Duke Math. J. 45 561–600.
  • [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.
  • [21] Ikeda, N., Nagasawa, M. and Watanabe, S. (1965). On branching Markov processes. Proc. Japan Acad. 41 816–821.
  • [22] Ikeda, N., Nagasawa, M. and Watanabe, S. (1968). Branching Markov processes. I. J. Math. Kyoto Univ. 8 233–278.
  • [23] Ikeda, N., Nagasawa, M. and Watanabe, S. (1968). Branching Markov processes. II. J. Math. Kyoto Univ. 8 365–410.
  • [24] Ikeda, N., Nagasawa, M. and Watanabe, S. (1969). Branching Markov processes. III. J. Math. Kyoto Univ. 9 95–160.
  • [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.
  • [26] Keiding, N. (1975). Extinction and exponential growth in random environments. Theoret. Population Biol. 8 49–63.
  • [27] Kulperger, R. (1979). Brillinger type mixing conditions for a simple branching diffusion process. Stochastic Process. Appl. 9 55–66.
  • [28] Kurtz, T. G. (1973). A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12 55–67.
  • [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.
  • [30] Kurtz, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3 29 pp. (electronic).
  • [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.
  • [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. To appear.
  • [33] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • [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).
  • [35] Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103–126.
  • [36] Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Teor. Verojatnost. i Primenen. 13 126–137.
  • [37] Li, Z. H. (1992). Measure-valued branching processes with immigration. Stochastic Process. Appl. 43 249–264.
  • [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.
  • [39] Li, Z. and Wang, Z. (1999). Measure-valued branching processes and immigration processes. Adv. Math. (China) 28 105–134.
  • [40] Mellein, B. (1982). Diffusion limits of conditioned critical Galton–Watson processes. Rev. Colombiana Mat. 16 125–140.
  • [41] Pakes, A. G. (1986). The Markov branching-catastrophe process. Stochastic Process. Appl. 23 1–33.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [47] Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 1–11 (electronic).
  • [48] Stannat, W. (2003). On transition semigroups of (A, Ψ)-superprocesses with immigration. Ann. Probab. 31 1377–1412.
  • [49] Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 141–167.