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Measure-Valued Branching Diffusions with Singular Interactions

Published online by Cambridge University Press:  20 November 2018

Steven N. Evans  and
Edwin A. Perkins
Steven N. Evans
Affiliation:
Department of Statistics University of California, Berkeley Berkeley, California 94720 U.S.A.
Edwin A. Perkins
Affiliation:
Department of Mathematics University of British Columbia 1984 Mathematics Road Vancouver, British Columbia V6T 1Y4
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Abstract

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The usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian particles in which the branching mechanism is critical. In this work we consider analogous processes that model the evolution of a system of two such populations in which there is inter-species competition or predation.

We first consider a competition model in which inter-species collisions may result in casualties on both sides. Using a Girsanov approach, we obtain existence and uniqueness of the appropriate martingale problem in one dimension. In two and three dimensions we establish existence only. However, we do show that, in three dimensions, any solution will not be absolutely continuous with respect to the law of two independent super-Brownian motions. Although the supports of two independent super-Brownian motions collide in dimensions four and five, we show that there is no solution to the martingale problem in these cases.

We next study a prédation model in which collisions only affect the "prey" species. Here we can show both existence and uniqueness in one, two and three dimensions. Again, there is no solution in four and five dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-process as stochastic integrals with respect to the related orthogonal martingale measure.

We also obtain existence and uniqueness for a related single population model in one dimension in which particles are killed at a rate proportional to the local density. This model appears as a limit of a rescaled contact process as the range of interaction goes to infinity.

Keywords

60G5760K3560J6060H1560G30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 1 , 01 February 1994 , pp. 120 - 168
Copyright
Copyright © Canadian Mathematical Society 1994

References

Barlow, M. T., Evans, S. N. and Perkins, E. A. (1991), Collision local times and measure-valued processes, Canad. J. Math. 43, 897–938.Google Scholar
Barlow, M. T. and Perkins, E. A. (1983), Strong existence, uniqueness and non-uniqueness in an equation involving local time, Séminaire de Probabilité XVII, Lect. Notes Math. 986, 32–61. (1993), in preparation.Google Scholar
Billingsley, P. (1968), Convergence of Probability Measures. Wiley, New York.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968), Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Dawson, D. A. (1978), Geostochastic calculus, Canad. J. Statist. 6, 143–168. Google Scholar
Dawson, D. A. (1991), Lecture notes on infinitely divisible random measures andsuperprocesses, In: Proc. 1990 Workshop on Stochastic Analysis and Related Topics, Silivri, Turkey, to appear.Google Scholar
Dawson, D. A., I. Iscoe and Perkins, E. A. (1989), Super-Brownian motion: path properties and hitting probabilities, Probab. Theor. Relat. Fields 83, 135–206.Google Scholar
Dawson, D. A. and Perkins, E. A. (1991), Historical Processes, Mem. Amer. Math. Soc. (454) 93, 1–179.Google Scholar
Dellacherie, C. and Meyer, P. A. (1978), Probability and Potential. North Holland Mathematical Studies 29, North Holland, Amsterdam.Google Scholar
Dellacherie, C. and Meyer, P. A. (1980), Probabilités et Potential: Théorie des Martingales, Hermann, Paris.Google Scholar
Dynkin, E. B. (1965), Markov Processes Vol I, Springer, Berlin. (1988), Representation for junctionals of superprocesses by multiple stochastic integrals with applications to self-intersection local times, Astérisque 157-158, 147–171.Google Scholar
Dynkin, E. B.(1989), Superprocesses and their linear additive Junctionals, Trans. Amer. Math. Soc. 316, 623–634.Google Scholar
Dynkin, E. B.(1993), On regularity of superprocesses,Vrob'db. Theor. Relat. Fields 95, 263–281.Google Scholar
Dynkin, E. B.(1992a), Superprocesses and partial differential equations, Ann. Probab., to appear.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986), Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Fitzsimmons, P. J. (1988), Construction and regularity of measure-valued Markov branching processes, Israel J. Math. 64, 337–361.Google Scholar
Fitzsimmons, P. J.(1992), On the martingale problem for measure-valued Markov branching processes, Seminar on Stochastic Processes 1991, Birkhauser, Boston, 39–51.Google Scholar
Hoover, D. N. and Keisler, H. J. (1984), Adapted probability distributions, Trans. Amer. Math. Soc 286, 159— 201.Google Scholar
Jacod, J. (1977), A general theorem of representation for martingales, In: Proceedings of Symposia in Pure Mathematics XXXI, American Mathematical Society, Providence, 37–54.Google Scholar
Konno, N. and Shiga, T. (1988), Stochastic differential equations jor some measure-valued diffusions, Probab. Theor. Relat. Fields 79, 201–225.Google Scholar
Le Gall, J. F. (1991), Brownian excursions, trees and measure-valued branching processes, Ann. Prob. 19, 1399–1439.Google Scholar
Le Gall, J. F. (1993),A class ofpath-valued Markov processes and its applications to superprocesses, Probab. Theor Relat. Fields 95, 25–46.Google Scholar
Mueller, C. and Perkins, E. A. (1992), The compact support property jor solutions to the heat equation with noise, Probab. Theor. Relat. Fields 93, 325–358.Google Scholar
Mueller, C. and Tribe, R. (1993), A stochastic PDE arising as the limit oja long range contact process, and its phase transitions, preprint.Google Scholar
Perkins, E. A. (1989), The Hausdorff measure of the closed support of super-Brownian motion, Ann. Inst. H. Poincaré 25, 205–224.Google Scholar
Perkins, E. A. (1989a), On a problem ojDurrett, handwritten manuscript.Google Scholar
Perkins, E. A. (1993), Measure-valued branching diffusions with spatial interactions, Probab. Theor. Relat. Fields 94, 189–245.Google Scholar
Perkins, E. A. and Taylor, S. J. (1993), On lower junctions for the local density oj super-Brownian motion, in preparation.Google Scholar
Reimers, M. (1989), One dimensional stochastic partial differential equations and the branching measure diffusion, Probab. Theor. Relat. Fields 81, 319–340.Google Scholar
Rogers, L. C. G. and D. Williams (1987), Diffusions, Markov Processes, and Martingales Vol. 2, Itô Calculus, Wiley, New York.Google Scholar
Rogers, L. C. G. and Walsh, J. B. (1991), Local time and stochastic area integrals, Ann. Probab. 19, 457–482.Google Scholar
Taylor, S. J. (1961), On the connections between generalized capacities and Hausdorff measures, Math. Proc. Cambridge Philos. Soc. 57, 524–531.Google Scholar
Walsh, J. B. (1986), An introduction to stochastic partial differential equations, École d'Été de Probabilités de Saint Flour XIV—1984, Lecture Notes in Math. 1180, Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar