A Continuous Super-Brownian Motion in a Super-Brownian Medium

Journal of Theoretical Probability

Volumn 10, Issue 1, 1997, Pages 213-276

  Dawson, Donald A ^a   Fleischmann, Klaus ^b  

^a FIELDS INSTITUTE   (Canada)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Branching rate functional; Catalyst process; Catalytic medium; Catalytic reaction diffusion equation; Critical branching; Measure valued branching; Occupation time; Random medium; Super Brownian motion; Superprocess

Indexed keywords

EID: 0031504086     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022606801625     Document Type: Article

References (41)

1. Barlow, M. T., Evans, S. N., and Perkins, E A. (1991). Collision local times and measure-valued processes. Can. J. Math. 43(5):897-938.
2. Bramson, M., and Neuhauser, C. (1992). A catalytic surface reaction model. J. Comput. Appl. Math. 40(2):157-161.
3. Barlow, M. T., and Perkins, E.A. (1994). On the filtration of historical Brownian motion. Ann. Prob. 22:1273-1294.
4. Chan, C. Y., and Fung, D. T. (1992). Dead cores and effectiveness of semilinear reaction diffusion systems. J. Math. Anal. Appl. 171(2):498-515.
5. Chadam, J. M., and Yin, H. M. (1994). A diffusion equation with localized chemical reactions. Proc. Roy. Soc. Edinburgh-Math. 37(1):101-118.
6. Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrsch. verw. Gebiete 40:125-145.
7. Dawson, D. A. (1993). Measure-valued Markov processes. In Hennequin, P. L. (Ed.), École d'été de probabilités de Saint Flour XXI-1991, Vol. 1541, pp. 1-260. Lecture Notes in Mathematics, Springer-Verlag, Berlin.
8. Delmas, J.-F. (1995). Super-mouvement brownien avec catalyse. CERMICS-ENPC, Noisy-Le-Grand. Preprint.
9. Dawson, D. A., and Fleischmann, K. (1988). Strong clumping of critical space-time branching models in subcritical dimensions. Stoch. Proc. Appl. 30:193-208.
10. Dawson, D. A., and Fleischmann, K. (1991). Critical branching in a highly fluctuating random mdium. Prob. Th. Rel. Fields 90:241-274.
11. Dawson, D. A., and Fleischmann, K. (1994). A super-Brownian motion with a single point catalyst. Stoch. Proc. Appl. 49:3-40.
12. Dawson, D. A., and Fleischmann, K. (1995). Super-Brownian motions in higher dimensions with absolutely continuous measure states. J. Th. Prob. 8(1):179-206.
13. Dawson, D. A., and Fleischmann, K. (1996). Longtime behavior of a branching process controlled by branching catalysts. WIAS Berlin. Preprint No. 261.
14. Dawson, D. A., Fleischmann, K., and LeGall, J.-F. (1995). Super-Brownian motions in catalytic media. In Heyde, C. C. (Ed.), Branching Processes: Proceedings of the First World Congress, Springer-Verlag, Vol. 99, Lectures Notes in Statistics, pp. 122-134.
15. Dawson, D. A., Fleischmann, K., and Leduc, G. (1996). Continuous dependence of a class of superprocesses on their branching parameters (in preparation).
16. Dawson, D. A., Fleischmann, K., Li, Y., and Mueller, C. (1995). Singularity of super-Brownian local time at a point catalyst. Ann. Prob. 23(1):37-55.
17. Dawson, D. A., Fleischmann, K., and Roelly, S. (1991). Absolute continuity for the measure states in a branching model with catalysts. In Stochastic Processes, Proc. Semin. Vancouver/CA 1990, Prog. Prob. 24:117-160.
18. Dawson, D. A., and Hochberg, K. J. (1979). The carrying dimension of a stochastic measure diffusion. Ann. Prob, 7:693-703.
19. Dynkin, E. B., and Kuznetsov, S. E. (1995). Superdiffusions and removable singularities for quasilinear partial differential equations. Comm. Pure and Appl. Math. 49:125-176.
20. Dawson, D., Li, Y., and Mueller, C. (1995). The support of measure-valued branching processes in a random environment. Ann. Prob. 23(4):1692-1718.
21. Dawson, D. A., Mueller, C., and Perkins, E. A. (1996). Mutually catalytic branching (in preparation).
22. Dawson, D. A., and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. No. 454.
23. Durrett, R., and Swindle, G. (1994). Coxistence results for catalysts. Prob. Th. Rel. Fields 98:489-515.
24. Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Prob. 19:1157-1194.
25. Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. Vol. 6 of CRM Monograph Series. Centre de Recherches Mathématiques, Université de Montréal, Amer. Math. Soc., Providence.
26. Dynkin, E. B. (1995). Branching with a single point catalyst. In Stochastic Analysis (Ithaca, NY, 1993), Vol. 57 of Proc. Sympos. Pure Math., pp. 423-425. Amer. Math. Soc., Providence, Rhodes Island.
27. Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
28. Evans, S. N., and Perkins, E. A. (1994). Measure-valued branching diffusions with singular interactions. Can. J. Math. 46(1):120-168.
29. Fleischmann, K., and LeGall, J.-F. (1995). A new approach to the single point catalytic super-Brownian motion. Prob. Th. Rel. Fields 102:63-82.
30. Fleischmann, K. (1994). Superprocesses in catalytic media. In Dawson, D. A. (Ed.), Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems, Vol. 5 CRM Proceedings & Lecture Notes, pp. 99-110. Centre de Recherches Mathématiques, Université de Montréal, Amer. Math. Soc., Providence.
31. Gorostiza, L. G., and Wakolbinger, A. (1993). Asymptotic behavior of a reaction-diffusion system. A probabilistic approach. Random and Comp. Dynamics 1:445-463.
32. Iscoe, I. (1986). A weighted occupation time for a class of measure-valued critical branching Brownian motions. Prob. Th. Rel. Fields 71:85-116.
33. Iscoe, I. (1988). On the supports of measure-valued critical branching Brownian motion. Ann. Prob. 16:200-221.
34. Konno, N., and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Prob. Th. Rel. Fields 79:201-225.
35. Leduc, G. (1995). A complete characterization of a class of branching measure-valued processes. Technical Report No. 275. Lab. Research Stat. Probab., Carleton Univ. Ottawa.
36. 2 in a domain. Prob. Th. Rel. Fields 102:393-432.
37. Manthey, R., and Mittmann, K. (1996). A growth estimate for continuous random fields. Math. Bohemica (to appear).
38. Reimers, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Prob. Th. Rel. Fields 81:319-340.
39. Revuz, D., and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin, Heidelberg, New York.
40. Schied, A. (1996). Sample path large deviations for super-Brownian motion. Prob. Th. Rel. Fields 104:319-347.
41. Sugitani, S. (1989). Some properties for the measure-valued branching diffusion process. J. Math. Soc. Japan 41(3):437-462.