Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:40:56.229Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness for a Competing Species Model

Published online by Cambridge University Press:  20 November 2018

Leonid Mytnik
Leonid Mytnik*
Affiliation:
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C. V6T 1Z2 email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that a martingale problem associated with a competing species model has a unique solution. The proof of uniqueness of the solution for the martingale problem is based on duality technique. It requires the construction of dual probability measures.

Keywords

60H1535R60stochastic partial differential equationmartingale problemduality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 2 , 01 April 1999 , pp. 372 - 448
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Adams, D. and Hedberg, L., Function Spaces and Potential Theory. Springer-Verlag, Berlin, 1996.Google Scholar
[2] Adler, R. and Mytnik, L., Bisexual branching diffusions. Adv. in Appl. Probab. 27 (1995), 980–1018.Google Scholar
[3] Aikawa, H., Potential theory, Part II. Lecture Notes in Math. 1633 (1996), 103–200.Google Scholar
[4] Baras, P. and Pierre, M., Problèmes paraboliques semi-linéaires avec données mesures. Appl. Anal. 18 (1984), 111–149.Google Scholar
[5] Barlow, M., Evans, S. and Perkins, E., Collision local times and measure-valued processes. Canad. J. Math. (5) 43 (1991), 897–938.Google Scholar
[6] Dawson, D., Geostochastic calculus. Canad. J. Statist. 6 (1978), 143–168.Google Scholar
[7] Dawson, D., Infinitely Divisible RandomMeasures and Superprocesses. In: Stochastic Analysis and Related Topics (Eds. Körezlioğlu, H. and Üstünel, A.), Birkhäuser, Boston, 1992.Google Scholar
[8] Dynkin, E., Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. Astérisque 157-158 (1988), 147–171.Google Scholar
[9] Dynkin, E., Superprocesses and partial differential equations. Ann. Probab. 21 (1993), 1185–1262.Google Scholar
[10] Ethier, S. N. and Kurtz, T. G., Markov Processes: Characterization and Convergence. John Wiley and Sons, New York, 1986.Google Scholar
[11] Evans, S. and Perkins, E., Absolute continuity results for superprocesses with some applications. Trans. Amer. Math. Soc. 325 (1991), 661–681.Google Scholar
[12] Evans, S. and Perkins, E., Measure-valued branching diffusions with singular interactions. Canad. J. Math. (1) 46 (1994), 120–168.Google Scholar
[13] Evans, S. and Perkins, E., Collision local times, historical calculus, and competing superprocesses. Preprint, 1997.Google Scholar
[14] Fleischmann, K., Critical behavior of some measure-valued processes. Math. Nachr. 135 (1988), 131–147.Google Scholar
[15] Jakubowski, A., On the Skorohod Topology. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), 263–285.Google Scholar
[16] Mitoma, I., Tightness of probabilities on C([0; 1]; S0) and D ([0; 1]; S0). Ann. Probab. 11 (1983), 989–999.Google Scholar
[17] Mytnik, L., Superprocesses in random environments. Ann. Probab. 24 (1996), 1953–1978.Google Scholar
[18] Mytnik, L., Weak uniqueness for the heat equation with noise. Ann. Probab. 26 (1998), 968–984.Google Scholar
[19] Perkins, E., On the Martingale Problem for Interactive Measure-Valued Branching Diffusions. Mem. Amer. Math. Soc. 549, 1995.Google Scholar
[20] Walsh, J., An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 (1986), 265–439.Google Scholar