Electronic Journal of Probability

Competing Species Superprocesses with Infinite Variance

Klaus Fleischmann and Leonid Mytnik

Full-text: Open access

Abstract

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 8, 59 p.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464037581

Digital Object Identifier
doi:10.1214/EJP.v8-136

Mathematical Reviews number (MathSciNet)
MR1986840

Zentralblatt MATH identifier
1065.60145

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
Superprocess with killing competing superprocesses interactive superprocesses superprocess with immigration measure-valued branching interactive branching state-dependent branching collision measure collision local time martingale problem

Citation

Fleischmann, Klaus; Mytnik, Leonid. Competing Species Superprocesses with Infinite Variance. Electron. J. Probab. 8 (2003), paper no. 8, 59 p. doi:10.1214/EJP.v8-136. https://projecteuclid.org/euclid.ejp/1464037581


Export citation

References

  • M.T. Barlow, S.N. Evans, and E.A. Perkins. Collision local times and measure-valued processes. Canad. J. Math., 43(5):897-938, 1991.
  • D.A. Dawson. Geostochastic calculus. Canadian J. Statistics, 6:143-168, 1978.
  • D.A. Dawson. Measure-valued Markov processes. In P.L. Hennequin, editor, École d'été de probabilités de Saint Flour XXI-1991, volume 1541 of Lecture Notes Math., pages 1-260. Springer-Verlag, Berlin, 1993.
  • D.A. Dawson, A.M. Etheridge, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Finite measure states. Ann. Probab., 30(4):1681-1762, 2002.
  • D.A. Dawson, A.M. Etheridge, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Infinite measure states. Electron. J. Probab., 7 (Paper no. 15) 61 pp., 2002.
  • D.A. Dawson and K. Fleischmann. Catalytic and mutually catalytic super-Brownian motions. In Ascona 1999 Conference. Volume 52 of Progress in Probability, pages 89-110. Birkhäuser Verlag, 2002.
  • D.A. Dawson, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Uniqueness. Ann. Inst. Henri Poincaré Probab. Statist. 39(1):135-191, 2003.
  • D.A. Dawson and E.A. Perkins. Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab., 26(3):1088-1138, 1998.
  • J.L. Doob. Measure Theory. Springer-Verlag, New York, 1994.
  • E.B. Dynkin. On regularity of superprocesses. Probab. Theory Related Fields, 95(2):263-281, 1993.
  • E.B. Dynkin. An Introduction to Branching Measure-valued Processes. American Mathematical Society, Providence, RI, 1994.
  • E.B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. Volume 50 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2002.
  • A.M. Etheridge. An Introduction to Superprocesses. Volume 20 of Univ. Lecture Series. AMS, Rhode Island, 2000.
  • S.N. Ethier and T.G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986.
  • S.N. Evans and E.A. Perkins. Measure-valued branching diffusions with singular interactions Canad. J. Math., 46(1):120-168, 1994.
  • S.N. Evans and E.A. Perkins. Collision local times, historical stochastic calculus, and competing superprocesses. Electron. J. Probab., 3 (Paper no. 5), 120 pp., 1998.
  • K. Fleischmann and J. Xiong. A cyclically catalytic super-Brownian motion. Ann. Probab., 29(2):820-861, 2001.
  • R.K. Getoor. On the construction of kernels. In Séminaires de probabilités IX. Volume 465 of Lecture Notes Math., pages 443-463. Springer Verlag, Berlin, 1974.
  • I. Iscoe. A weighted occupation time for a class of measure-valued critical branching Brownian motions. Probab. Theory Related Fields, 71:85-116, 1986.
  • O. Kallenberg. Foundations of Modern Probability. Springer-Verlag, New York, 1997.
  • J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser Verlag, Basel, 1999.
  • C. Mueller and E.A. Perkins. Extinction for two parabolic stochastic PDE's on the lattice. Ann. Inst. H. Poincaré Probab. Statist., 36(3):301-338, 2000.
  • L. Mytnik. Collision measure and collision local time for ($\alpha, d, \beta$) superprocesses. J. Theoret. Probab., 11(3):733-763, 1998.
  • L. Mytnik. Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields, 112(2):245-253, 1998.
  • L. Mytnik. Uniqueness for a competing species model. Canad. J. Math., 51(2):372-448, 1999.
  • E.A. Perkins. On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc., 549, 1995.
  • E.A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In École d'été de probabilités de Saint Flour XXIX-1999, Lecture Notes Math., pages 125-324, Springer-Verlag, Berlin, 2002.
  • Ph. Protter. Stochastic Integration and Differential Equations, a New Approach. Volume 21 of Appl. Math., Springer-Verlag, Berlin, 1990.
  • S. Roelly-Coppoletta. A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics, 17:43-65, 1986.
  • J.B. Walsh. An introduction to stochastic partial differential equations. Volume 1180 of Lecture Notes Math., pages 266-439. École d'été de probabilités de Saint-Flour XIV - 1984, Springer-Verlag Berlin, 1986.
  • K. Yosida. Functional Analysis. Springer-Verlag, 4th edition, 1974.