The Annals of Probability

Mutually catalytic branching in the plane: Finite measure states

Donald A. Dawson, Alison M. Etheridge, Klaus Fleischmann, Leonid Mytnik, Edwin A. Perkins, and Jie Xiong

Full-text: Open access

Abstract

We study a pair of populations in $\mathbb{R}^{2}$ which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding $\mathbb{Z}^{2}$-lattice model studied by D. A. Dawson and E. A. Perkins and resolves the large scale mass-time-space behavior of that model.

Article information

Source
Ann. Probab., Volume 30, Number 4 (2002), 1681-1762.

Dates
First available in Project Euclid: 10 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039548370

Digital Object Identifier
doi:10.1214/aop/1039548370

Mathematical Reviews number (MathSciNet)
MR1944004

Zentralblatt MATH identifier
1017.60098

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Catalytic super-Brownian motion catalytic super-random walk collision local time duality superprocesses martingale problem segregation of types stochastic PDE

Citation

Dawson, Donald A.; Etheridge, Alison M.; Fleischmann, Klaus; Mytnik, Leonid; Perkins, Edwin A.; Xiong, Jie. Mutually catalytic branching in the plane: Finite measure states. Ann. Probab. 30 (2002), no. 4, 1681--1762. doi:10.1214/aop/1039548370. https://projecteuclid.org/euclid.aop/1039548370


Export citation

References

  • [1] ATHREy A, S. and TRIBE, R. (2000). Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. Ann. Probab. 28 1711-1734.
  • [2] BARLOW, M. T., EVANS, S. N. and PERKINS, E. A. (1991). Collision local times and measure-valued processes. Canad. J. Math. 43 897-938.
  • [3] BARLOW, M. T. and PERKINS, E. A. (1994). On the filtration of historical Brownian motion. Ann. Probab. 22 1273-1294.
  • [4] BASS, R. F. and PERKINS, E. A. (2003). Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc. 335 373-405.
  • [5] BILLINGSLEY, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • [6] COX, J. T., DAWSON, D. A. and GREVEN, A. (2002). Mutually cataly tic super branching random walks: Large finite sy stems and renormalization analysis. Preprint.
  • [7] COX, T., KLENKE, A. and PERKINS, E. A. (2000). Convergence to equilibrium and linear sy stem duality. CMS Conf. Proc. 26 41-66.
  • [8] DAWSON, D. A., ETHERIDGE, A. M., FLEISCHMANN, K., My TNIK, L., PERKINS, E. A.
  • and XIONG, J. (2002). Mutually cataly tic branching in the plane: Infinite measure states. Electron. J. Probab. 7.
  • [9] DAWSON, D. A. and FLEISCHMANN, K. (1995). Super-Brownian motions in higher dimensions with absolutely continuous measure states. J. Theoret. Probab. 8 179-206.
  • [10] DAWSON, D. A. and FLEISCHMANN, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 213-276.
  • [11] DAWSON, D. A. and FLEISCHMANN, K. (1997). Longtime behavior of a branching process controlled by branching cataly sts. Stochastic Process. Appl. 71 241-257.
  • [12] DAWSON, D. A., FLEISCHMANN, K., My TNIK, L. T., PERKINS, E. A. and XIONG, J.
  • (2002). Mutually cataly tic branching in the plane: Uniqueness. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [13] DAWSON, D. A. and MARCH, P. (1995). Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems. J. Funct. Anal. 132 417-472.
  • [14] DAWSON, D. A. and PERKINS, E. A. (1998). Long-time behaviour and coexistence in a mutually cataly tic branching model. Ann. Probab. 26 1088-1138.
  • [15] DAWSON, D. A. and PERKINS, E. A. (1999). Measure-valued processes and renormalization of branching particle sy stems. In Stochastic Partial Differential Equations: Six Perspectives (R. Carmona and B. Rozovskii, eds.) 45-106. Amer. Math. Soc., Providence, RI.
  • [16] DELMAS, J.-F. (1996). Super-mouvement brownien avec cataly se. Stochastics Stochastics Rep. 58 303-347.
  • [17] DELMAS, J.-F. and FLEISCHMANN, K. (2001). On the hot spots of a cataly tic super-Brownian motion. Probab. Theory Related Fields 121 389-421.
  • [18] DONNELLY, P. and KURTZ, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166-205.
  • [19] EIGEN, M. (1971). Selforganization of matter and the evolution of biological macromolecules. Die Naturwissenschaften 58 465-523.
  • [20] EIGEN, M. (1982). The Hy percy cle: Principle of Natural Selforganization. Springer, Berlin.
  • [21] ETHERIDGE, A. M. and FLEISCHMANN, K. (1998). Persistence of a two-dimensional superBrownian motion in a cataly tic medium. Probab. Theory Related Fields 110 1-12.
  • [22] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [23] EVANS, S. N. and PERKINS, E. A. (1994). Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 120-168.
  • [24] FELLER, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
  • [25] FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • [26] FLEISCHMANN, K. and KLENKE, A. (1999). Smooth density field of cataly tic super-Brownian motion. Ann. Appl. Probab. 9 298-318.
  • [27] FLEISCHMANN, K. and KLENKE, A. (2000). The biodiversity of cataly tic super-Brownian motion. Ann. Appl. Probab. 10 1121-1136.
  • [28] FLEISCHMANN, K. and XIONG, J. (2001). A cy clically cataly tic super-Brownian motion. Ann. Probab. 29 820-861.
  • [29] JACOD, J. and SHIRy AEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [30] JAKUBOWSKI, A. (1986). On the Skorohod topology. Ann. Inst. H. Poincaré Ser. B 22 263-285.
  • [31] KONNO, N. and SHIGA, T. (1988). Stochastic differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201-225.
  • [32] MEy ER, P.-A. (1966). Probability and Potentials. Blaisdell, Toronto.
  • [33] MITOMA, I. (1985). An -dimensional inhomogeneous Langevin equation. J. Funct. Anal. 61 342-359.
  • [34] My TNIK, L. (1996). Superprocesses in random environments. Ann. Probab. 24 1953-1978.
  • [35] My TNIK, L. (1998). Uniqueness for a mutually cataly tic branching model. Probab. Theory Related Fields 112 245-253.
  • [36] PERKINS, E. A. (1995). On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 1-89.
  • [37] PERKINS, E. A. (2000). Dawson-Watanabe superprocesses and measure-valued diffusions. École d'Été de Probabilités de Saint-Flour 1999. Lecture Notes in Math. 1781 125-329. Springer, New York.
  • [38] REIMERS, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 319-340.
  • [39] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [40] RUDIN, W. (1974). Real and Complex Analy sis, 2nd ed. McGraw-Hill, New York.
  • [41] SHIGA, T. (1988). Stepping stone models in population genetics and population dy namics. In Stochastic Processes in physics and Engineering (S. Albeverio et al., eds.) 345-355. Reidel, Dordrecht.
  • [42] SHIGA, T. (1994). Two contrasting properties of solutions for one-dimensional stochastic pde's. Canad. J. Math. 46 415-437.
  • [43] SHIGA, T. and SHIMIZU, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Ky oto Univ. 20 395-416.
  • [44] SPITZER, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.
  • [45] WALSH, J. B. (1986). An introduction to stochastic partial differential equations. École d'Été de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 266-439. Springer, Berlin.
  • KNOXVILLE, TENNESSEE 37996-1300 E-MAIL: jxiong@math.utk.edu