Competing Species Superprocesses with Infinite Variance
Leonid Mytnik (Technion - Israel Institute of Technology)
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.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1 - 59
Publication Date: May 22, 2003
DOI: 10.1214/EJP.v8-136
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. Math. Review 93a:60119
- D.A. Dawson. Geostochastic calculus. Canadian J. Statistics, 6:143-168, 1978. Math. Review 81g:60076
- D.A. Dawson. Measure-valued Markov processes. In P.L. Hennequin, editor, â¦cole d'Ãtà de probabilitÃs de Saint Flour XXI-1991, volume 1541 of Lecture Notes Math., pages 1-260. Springer-Verlag, Berlin, 1993. Math. Review 94m:60101
- 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. Math. Review 1 944 004
- 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. (electronic), 2002. Math. Review 1 921 744
- 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. Birkhâ°user Verlag, 2002. Math. Review 1 958 811
- D.A. Dawson, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Uniqueness. Ann. Inst. Henri Poincarà Probab. Statist. 39(1):135-191, 2003. Math. Review 1 959 845
- 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. Math. Review 99f:60167
- J.L. Doob. Measure Theory. Springer-Verlag, New York, 1994. Math. Review 95c:28001
- E.B. Dynkin. On regularity of superprocesses. Probab. Theory Related Fields, 95(2):263-281, 1993. Math. Review 94f:60107
- E.B. Dynkin. An Introduction to Branching Measure-valued Processes. American Mathematical Society, Providence, RI, 1994. Math. Review 96f:60145
- E.B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. Volume 50 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2002. Math. Review 2003c:60001
- A.M. Etheridge. An Introduction to Superprocesses. Volume 20 of Univ. Lecture Series. AMS, Rhode Island, 2000. Math. Review 2001m:60111
- S.N. Ethier and T.G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. Math. Review 88a:60130
- S.N. Evans and E.A. Perkins. Measure-valued branching diffusions with singular interactions Canad. J. Math., 46(1):120-168, 1994. Math. Review 94J:60099
- 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. (electronic), 1998. Math. Review 99h:60098
- K. Fleischmann and J. Xiong. A cyclically catalytic super-Brownian motion. Ann. Probab., 29(2):820-861, 2001. Math. Review 2002h:60224
- R.K. Getoor. On the construction of kernels. In SÃminaires de probabilitÃs IX. Volume 465 of Lecture Notes Math., pages 443-463. Springer Verlag, Berlin, 1974. Math. Review 55:9289
- I. Iscoe. A weighted occupation time for a class of measure-valued critical branching Brownian motions. Probab. Theory Related Fields, 71:85-116, 1986. Math. Review 87c:60070
- O. Kallenberg. Foundations of Modern Probability. Springer-Verlag, New York, 1997. Math. Review 99e:60001
- J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhâ°user Verlag, Basel, 1999. Math. Review 2001g:60211
- C. Mueller and E.A. Perkins. Extinction for two parabolic stochastic PDE's on the lattice. Ann. Inst. H. Poincarà Probab. Statist., 36(3):301-338, 2000. Math. Review 2001i:60104
- L. Mytnik. Collision measure and collision local time for (alpha,d,beta) superprocesses. J. Theoret. Probab., 11(3):733-763, 1998. Math. Review 2000a:60146
- L. Mytnik. Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields, 112(2):245-253, 1998. Math. Review 99i:60125
- L. Mytnik. Uniqueness for a competing species model. Canad. J. Math., 51(2):372-448, 1999. Math. Review 2000g:60112
- E.A. Perkins. On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc., 549, 1995. Math. Review 95i:60076
- E.A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In â¦cole d'Ãtà de probabilitÃs de Saint Flour XXIX-1999, Lecture Notes Math., pages 125-324, Springer-Verlag, Berlin, 2002. Math. Review 1 915 445
- Ph. Protter. Stochastic Integration and Differential Equations, a New Approach. Volume 21 of Appl. Math., Springer-Verlag, Berlin, 1990. Math. Review 91i:60148
- S. Roelly-Coppoletta. A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics, 17:43-65, 1986. Math. Review 88i:60132
- J.B. Walsh. An introduction to stochastic partial differential equations. Volume 1180 of Lecture Notes Math., pages 266-439. â¦cole d'Ãtà de probabilitÃs de Saint-Flour XIV - 1984, Springer-Verlag Berlin, 1986. Math. Review 88a:60114
- K. Yosida. Functional Analysis. Springer-Verlag, 4th edition, 1974. Math. Review 50:2851

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