## The Annals of Probability

### Particle Representations for Measure-Valued Population Models

#### Abstract

Models of populations in which a type or location, represented by a point in a metric space$E$, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measure-valued processes are obtained as infinite population limits for a large class of neutral population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population and the de Finetti measures associated with an $E^{\infty}$ -valued particle model$X=(X_1, X_2\ldots)$ such that, for each $t \geq 0,(X_1(t),X_2(t),\ldots)$ is exchangeable. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both neutral Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of the Dawson-Perkins historical process and Perkins’s historical stochastic integral can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behavior are described.

#### Article information

Source
Ann. Probab. Volume 27, Number 1 (1999), 166-205.

Dates
First available in Project Euclid: 29 May 2002

http://projecteuclid.org/euclid.aop/1022677258

Digital Object Identifier
doi:10.1214/aop/1022677258

Mathematical Reviews number (MathSciNet)
MR1681126

Zentralblatt MATH identifier
0956.60081

#### Citation

Donnelly, Peter; Kurtz, Thomas G. Particle Representations for Measure-Valued Population Models. Ann. Probab. 27 (1999), no. 1, 166--205. doi:10.1214/aop/1022677258. http://projecteuclid.org/euclid.aop/1022677258.

#### References

• AVRAM, F. 1988. Weak convergence of the variations, iterated integrals, and Doleans Dade ´ exponentials of sequences of semimartingales. Ann. Probab. 16 246 250.
• BHATT, A. G. and KARANDIKAR, R. L. 1993. Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21 2246 2268.
• BLACKWELL, D. and DUBINS, L. E. 1983. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 691 692.
• DAWSON, D. A. 1993. Measure-valued Markov processes. Ecole d'Ete de Probabilites de Saint´ ´ Flour XXI. Lecture Notes in Math. 1541. Springer, Berlin.Z.
• DAWSON, D. A. and PERKINS, E. A. 1991. Historical processes. Mem. Amer. Math. Soc. 93 454.
• DONNELLY, P. and KURTZ, T. G. 1996. A countable representation of the Fleming Viot measure-valued diffusion. Ann. Probab. 24 698 742.
• DYNKIN, E. B. 1965. Markov Processes I. Springer, Berlin.
• EL KAROUI, N. and ROELLY, S. 1991. Proprietes de martingales, explosion et representation de Levy Khinchine d'une classe du processus de branchement a valeurs mesures. ´ Stochastic Process. Appl. 38 239 266.
• ETHERIDGE, A. and MARCH, P. 1991. A note on superprocesses. Probab. Theory Related Fields 89 141 147.
• ETHIER, S. N. and KURTZ, T. G. 1986. Markov Processes: Characterization and Convergence. Wiley, New York.
• ETHIER, S. N. and KURTZ, T. G. 1993. Fleming Viot processes in population genetics. SIAM J. Control Optim. 31 345 386.
• EVANS, S. N. 1993. Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959 971.
• EVANS, S. N. and PERKINS, E. 1990. Measure-valued Markov branching processes conditioned on non-extinction. Israel J. Math. 71 329 337.
• KINGMAN, J. F. C. 1982. The coalescent. Stochastic Process. Appl. 13 235 248.
• KURTZ, T. G. 1998. Martingale problems for conditional distributions of Markov processes. Elec. J. Probab. 3.
• KURTZ, T. G. and PROTTER, P. 1991. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035 1070.
• LOEVE, M. 1963. Probability Theory, 3rd ed. Van Nostrand, Princeton.
• PERKINS, E. A. 1991. Conditional Dawson Watanabe processes and Fleming Viot processes. In Seminar on Stochastic Processes 142 155. Birkhauser, Boston. ¨
• PERKINS, E. A. 1992. Measure-valued branching diffusion with spatial interactions. Probab. Theory Related Fields 94 189 245.
• PERKINS, E. A. 1995. On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 1 89.
• PITMAN, J. 1997. Coalescents with multiple collisions. Preprint.
• TRIBE, R. 1992. The behavior of superprocesses near extinction. Ann. Probab. 20 286 311.
• UNITED KINGDOM MADISON, WISCONSIN 53706-1388 E-MAIL: donnelly@stats.ox.ac.uk E-MAIL: kurtz@math.wisc.edu