Mutually catalytic super branching random walks: Large finite systems and renormalization analysis

Memoirs of the American Mathematical Society

Volumn 171, Issue 809, 2004, Pages

  Cox, J T ^a   Dawson, D A ^b   Greven, A ^c  

^a SYRACUSE UNIVERSITY   (United States)

^b CARLETON UNIVERSITY   (Canada)

^c UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Cluster formation; Continuum limit; Finite systems; Mutually catalytic branching; Renormalization; Spatial branching processes

Indexed keywords

EID: 3943107464     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0809     Document Type: Article

References (42)

1. [Aldous] D. Aldous. Exchangeability and Related Topics, Lecture Notes in Math. 1117, 1-198 (1985), Springer-Verlag.
2. [BCGH] J.B. Baillon, Ph. Clément, A. Greven, F. den Hollander, On the attracting orbit of a nonlinear transformation arising from normalization of hierarchically interacting diffusions Part 2: The non-compact case, Journal of Functional Analysis, Vol. 146, No. 1, 236-298 (1997)
3. [CFG] J. T. Cox, K. Fleischmann and A. Greven, Comparison of interacting diffusions and an application to their ergodic theory Probab.Theory Rel. Fields 105(4), 513-528 (1996).
4. [CG1] J. T. Cox, A. Greven, The longtime behavior of some finite particle systems, Prob. Theory and Rel. Fields 85, 195-237 (1990).
5. [CG2] J. T. Cox, A. Greven, The finite system scheme: An abstract theorem and a new example, CRM Proceedings and Lecture Notes, Vol. 5, 55-67 (1994).
6. [CG3] J. T. Cox, A. Greven, Ergodic theorems for systems of locally interacting diffusions. Ann. of Prob. 22(2). 833-853 (1994).
7. [CGS1] J. T. Cox, A. Greven, T. Shiga, Finite and infinite systems of interacting diffusions, Prob. Theory and Rel. Fields 102, 165-197 (1995).
8. [CGS2] J. T. Cox, A. Greven, T. Shiga, Finite and infinite systems of interacting diffusions, Part II, Math. Nachrichten 192 (1998).
9. [CGr] J. T. Cox, D. Griffeath: Diffusive clustering in the two-dimensional voter model Ann. Probab. 14(2), 347-370 (1986).
10. [CKP] J. T. Cox, A. Klenke, E. Perkins, Convergence to Equilibrium and Linear Systems Proceedings of the Inter. Conf. on Stochastic Models, Conference Proceedings Series of the Canadian Math. Society, Vol. 26, 41-66 (2000).
11. [D] D.A. Dawson, Measure valued Markov processes, In École d'été de Probabilités de Saint Flour XXI, Notes in Mathematics 1541, 1-261, Springer-Verlag (1993).
12. [DEFMPX1] 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. Vol. 30, 1681-1762 (2002).
13. [DEFMPX2] D.A. Dawson, A.M. Etheridge, K. Fleischmann, L. Mytnik, E.A. Perkins and J. Xiong, Mutually catalytic branching in the plane: In-finite measure states, Electron. J. Probab. Vol. 7, Paper 15 (2002).
14. [DFMPX] D.A. Dawson, K. Fleischmann, L. Mytnik, E.A. Perkins and J. Xiong, Mutually catalytic branching in the plane: Uniqueness, Ann. l'Institut Henri Poincaré, Probab. Statist. Vol 39, 135-191 (2003).
15. [DF1] D.A. Dawson, K. Fleischmann, A continuous super-Brownian motion in a super-Brownian medium, J. Theoret. Probab., 10(1), 213-276 (1997).
16. [DF2] D.A. Dawson, K. Fleischmann, Longtime behavior of a branching process controlled by branching catalysts, Stochastic Process. Appl. 71(2), 241-257 (1997).
17. [DG1] D.A. Dawson, A. Greven, Multiple Scale Analysis of Interacting Diffusions, Prob. Theory and Rel. Fields 95, 467-508 (1993).
18. [DG2] D.A. Dawson, A. Greven, Hierarchical models of interacting diffusions: Multiple time scales, Phase transitions and cluster formation, Prof. Theory and Rel. Fields 96, 435-473 (1993).
19. [DG3] D.A. Dawson, A. Greven, Multiple space-time scale analysis for interacting branching models. Electronic Journal of Probability, Vol. 1, Paper no. 14 (1996). (http://www.emis.de/journals/EJP-ECP/ejp1contents.html)
20. [DG4] D.A. Dawson, A. Greven, State dependent multi-type spatial branching processes and their longtime behavior, Preprint (2001).
21. [DG5] D.A. Dawson, A. Greven, Renormalization analysis of the historical process of state dependent branching system, manuscript (2002).
22. [DGV] D. A. Dawson, A. Greven, J. Vaillancourt, Equilibria and Quasi-equilibria for infinite systems of Fleming- Viot processes. Transactions of the American Math. Society, Vol. 347, No. 7 (1995).
23. [DGW] D.A. Dawson, L.G. Gorostiza and A. Wakolbinger. Occupation time fluctuations in branching systems, J. Theoretical Probab. 14, 729-796 (2001).
24. [DM] D. A. Dawson and P. March, Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems, J. Funct. Anal., Vol. 132, 417-472 (1995).
25. [DP] D. A. Dawson and E. Perkins, (1998). Longtime behavior and coexistence in a mutually catalytic branching model, Ann. Prob. Vol 26, No. 3, 1088-1138 (1995).
26. [EK] S. N. Ethier and T. G. Kurtz, Markov Processes - Characterization and Convergence John Wiley and Sons (1986).
27. [EF] S. Evans and K. Fleischmann, Cluster formation in a stepping stone model with continuous hierarchically structured sites , Ann. Prob. Vol. 24, 1926-1952 (1996).
28. [FG] K. Fleischmann, A. Greven, Time-space analysis of the cluster-formation in interacting diffusions, Electron. J. Probab. Vol. 1, Paper 6 (1996).
29. [FK] K. Fleischmann, A. Klenke, Smooth density field of catalytic super Brownian motion. Ann. Appl. Probab. 9(2), 343-363 (1999).
30. [G1] A. Greven, Renormalization Analysis of interacting spatial stochastic systems, Proceedings of the Royal Dutch Academy, 171-196 (2000).
31. [G2] A. Greven, Interacting stochastic systems: Longtime behavior and its renormalization analysis, Jber. d. Dg. Math. Verein. Vol. 102 (2000).
32. [GKW1] A. Greven, A. Klenke, A. Wakolbinger, The longtime behavior of branching random walk in a catalytic medium, EJP, Vol. 4, paper 12, 80, pages (electronic), (1999).
33. [GKW2] A. Greven, A. Klenke, A. Wakolbinger, Interacting Fisher-Wright Diffusions in a Catalytic Medium, Prob. Theory and Rel. Fields, Vol. 120, No. 1, 85-117 (2001).
34. [GKW3] A. Greven, A. Klenke, A. Wakolbinger, Interacting diffusions in a random medium: comparison and longtime behavior, Stochastic Process. Appl., Vol. 98, 23-41 (2002).
35. [HS], Hollander, F. den, J. M. Swart: Renormalization of hierarchically interacting Isotropic diffusions, J. Stat. Phys. 93, 243-291 (1998).
36. [K] A. Klenke, Different clustering regimes in systems of hierarchically interacting diffusions, Ann. Probab. 24(2), 660-697 (1996).
37. [LS] T.M. Liggett and F. Spitzer. Ergodic Theorems for coupled random walks and other systems with locally interacting components, Z. Wahr. V. Geb. 56(4), 443-468 (1981).
38. [M] L. Mytnik, Uniqueness for a mutually catalytic branching model, Prob. Theory Rel. Fields 112, 245-253 (1999).
39. [R] H. Rost, Darstellung einer Ordnung von Massen durch Stoppzeiten, ZfW 15, 19-28 (1970).
40. [W] A. Winter, Multiple scale analysis of branching processes under the Palm distribution, Elect. J. Probab. Vol. 7, Paper 13 (2002).
41. [Z1] I. Zähle, Renormalization of the Voter Model in Equilibrium. Ann. of Probability, Vol. 7 (2001).
42. [Z2] I. Zähle, Renormalizations of branching random walks in equilibrium, Elect. J. Probab. Vol. 7, Paper 7 (2002).