Measure-Valued Branching Diffusions with Singular Interactions

Evans, Steven N. ; Perkins, Edwin A.

Canadian journal of mathematics, Tome 46 (1994) no. 1, pp. 120-168

Résumé

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.

Détail

DOI : 10.4153/CJM-1994-004-6

Mots-clés : 60G57, 60K35, 60J60, 60H15, 60G30

Bibliographie

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