Collision Local Times, Historical Stochastic Calculus, and Competing Species

Steven N. Evans (University of California at Berkeley)
Edwin A. Perkins (The University of British Columbia)

Abstract


Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-120

Publication Date: April 8, 1998

DOI: 10.1214/EJP.v3-27

References

  1. R. J. Adler and R. Tribe, Uniqueness of a historical SDE with a singular interaction, J. Theoret. Probab., 1997. In Press. Math Review article not available.
  2. M. T. Barlow, S. N. Evans, and E. A. Perkins, Collision local times and measure-valued processes. Canad. J. Math., 43:897-938, 1991. Math Review link
  3. B. Bolker and S. W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoret. Popul. Biol., 1997. In Press. Math Review article not available.
  4. B. Bolker, S. W. Pacala, and S. A. Levin, Moment methods for stochastic processes in continuous space and time. In U. Dieckmann and J. Metz, editors, Low-Dimensional Dynamics of Spatial Ecological Systems, 1997. In Press. Math Review article not available.
  5. D. A. Dawson, Measure-valued Markov processes, In Ecole d'Ete de Probabilites de Saint-Flour XXI-1991, volume 1541 of Lecture Notes in Math., pages 1-260. Springer, Berlin, 1993. Math Review link
  6. D. A. Dawson and K. Fleischmann, Longtime behavior of branching processes controlled by branching catalysts. Preprint, 1996. Math Review article not available.
  7. D. A. Dawson and K. Fleischmann, A continuous super-Brownian motion in a super-Brownian medium, J. Theoret. Probab., 10: 213-276, 1997. Math Review link
  8. D. A. Dawson and P. March, Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems, J. Funct. Anal., 132: 417-472, 1995. Math Review link
  9. D. A. Dawson and E. A. Perkins, Historical processes, Mem. Amer. Math. Soc., 93, 1991. Math Review link
  10. D. A. Dawson and E. A. Perkins, Long-time behaviour and co-existence in a mutually catalytic branching model. Preprint, 1997. To appear in Ann Probab. Math Review article not available.
  11. C. Dellacherie and P.-A. Meyer, Probabilities and Potential, Number 29 in Mathematics Studies. North-Holland, Amsterdam, 1978. Math Review link
  12. C. Dellacherie and P.-A. Meyer, Probabilities and Potential B, Number 72 in Mathematics Studies. North-Holland, Amsterdam, 1982. Math Review link
  13. E. B. Dynkin, Three classes of infinite dimensional diffusions, J. Funct. Anal., 86:75-110, 1989. Math Review link
  14. E. B. Dynkin, Path processes and historical processes, Probab. Theory Relat. Fields, 90:89-115, 1991. Math Review link
  15. E. B. Dynkin, On regularity of superprocesses, Probab. Theory Relat. Fields, 95:263-281, 1993. Math Review link
  16. N. El-Karoui and S. Roelly, Proprietes de martingales, explosion et representation de Levy-Khintchine d'une class de processus de branchement a valeurs mesures. Stoch. Proc. Appl., 39: 239-266, 1991. Math Review link
  17. A. M. Etheridge and K. Fleischmann, Persistence of a two-dimensional super-Brownian motion in a catalytic medium, Preprint, 1997. Math Review article not available.
  18. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. Math Review link
  19. S. N. Evans and E. A. Perkins, Absolute continuity results for superprocesses with some applications, Trans. Amer. Math. Soc., 325: 661-681, 1991. Math Review link
  20. S. N. Evans and E. A. Perkins, Measure-valued branching diffusions with singular interactions, Canad. J. Math., 46: 120-168, 1994. Math Review link
  21. S. N. Evans and E. A. Perkins, Explicit stochastic integral representations for historical functionals, Ann. Prob., 23: 1772-1815, 1995. Math Review link
  22. P. J. Fitzsimmons, Construction and regularity of measure-valued Markov branching processes, Israel J. Math., 64: 337-361, 1988. Math Review link
  23. K. Fleischmann and A. Klenke, Smooth density field of catalytic super-Brownian motion, Preprint, 1997. Math Review article not available.
  24. D. N. Hoover and H. J. Keisler, Adapted probability distributions, Trans. Amer. Math. Soc., 286: 159-201, 1984. Math Review link
  25. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. Math Review link
  26. M. Lopez, Path properties and convergence of interacting superprocesses, PhD thesis, University of British Columbia, 1996. Math Review article not available.
  27. L. Mytnik, Uniqueness of a competing species model, Preprint, 1997. Math Review article not available.
  28. C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, Preprint, 1997. Math Review article not available.
  29. L. Overbeck, Nonlinear superprocesses, Ann. Probab., 24: 743-760, 1996. Math Review link
  30. E. A. Perkins, Measure-valued branching diffusion with spatial interactions, Prob. Theory Relat. Fields, 94: 189-245, 1992. Math Review link
  31. E. A. Perkins, On the martingale problem for interactive measure-valued branching diffusions, Mem. Amer. Math. Soc., 115, 1995. Math Review link
  32. J. B. Walsh, An introduction to stochastic partial differential equations. In Ecole d'Ete de Probabilites de Saint-Flour XIV-1984, volume 1180 of Lecture Notes in Math., pages 265-439, Springer, Berlin, 1986. Math Review link
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.