Asymptotic fluctuations and critical dimension for a two-level branching system

Bernoulli

Volumn 2, Issue 2, 1996, Pages 109-132

  Gorostiza, Luis G ^a  

^a Centro de Investigation y de Estudios Amnzados   (Mexico)

Author keywords

Asymptotic fluctuations; Critical dimension; Two level branching system

Indexed keywords

EID: 0011294075     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/bj/1193839219     Document Type: Article

References (50)

1. Aldous, D. (1989) Stopping times and tightness II. Ann. Probab17, 586-595.
2. Blumenthal, R.M. and Getoor, R.K. (1968) Markov Processes and Potential Theory, New York: Academic Press.
3. Bojdecki, T. and Gorostiza, L.G. (1986) Langevin equations for S'-valued Gaussian processes and fluctuation limits of infinite particle systems. Probab. Theory Related Fields, 73, 227-244.
4. Bojdecki, T. and Gorostiza, L.G. (1995) Self-intersection local time for Gaussian S’fR'^-processes: Existence, path continuity and examples. Stochastic Process, AppL, 60, 191-226.
5. Bojdecki, T. and Jakubowski, J. (1990) Stochastic integration for inhomogeneous Wiener process in the dual of a nuclear space. J. Multivariate Anal., 34, 185-210.
6. Cremers, H. and Kadelka, D. (1986) On weak convergence of integral functions of stochastic processes with applications to processes taking values in Lf. Stochastic Process. AppL, 21, 305-317.
7. Dawson, D.A. (1993) Measure-valued Markov processes. In Ecole d’Elé de Probabilités de Saint-Flour XXI-I991, pp. 1-260. Lecture Notes in Math. 1541, New York: Springer-Verlag.
8. Dawson, D.A. and Gartner, J. (1995) Multilevel large deviations and interacting diffusions II. Preprint.
9. Dawson, D.A. and Gorostiza, L.G. (1989) Generalized solutions of stochastic evolution equations. In G. Da Prato and L. Tubaro (eds), Stochastic Partial Differential Equations and Applications 11, pp. 53-65. Lecture Notes in Math. 1930. New York: Springer-Verlag.
10. Dawson, D.A. and Gorostiza, L.G. (1990) Generalized solutions of a class of nuclear space-valued stochastic evolution equations. AppL Math. Optim., 22, 241-263.
11. Dawson, D.A. and Hochberg, K.J. (1991) A multilevel branching model. Adv. Appl. Probab., 23,701-715.
12. Dawson, D.A. and Perkins, E.A. (1991) Historical processes. Mem. Amer. Math. Soc. 454. Providence, RI: American Mathematical Society.
13. Dawson, D.A. and Wu, Y. (1995) Multilevel multi type branching models of an information system. In K. Athreya and P. Jagers (eds), Proceedings of IMA Workshop on Branching Processes. To appear.
14. Dawson, D.A., Hochberg, K.J. and Wu, Y. (1990) Multilevel branching systems. In T. Hida, H.H. Kuo, J. Potthof and L. Streit (eds), White Noise Analysis: Mathematics and Applications, pp. 93-107. Singapore; World Scientific.
15. Dawson, D.A., Hochberg, K.J. and Vinogradov, V. (1994) On path properties of super-2 processes I. In D.A. Dawson (ed,), Measure-valued Processes, Stochastic Partial Differential Equations, and Interacting Systems, pp. 69-82. CRM Proc. Lecture Notes 5. Providence, RI: American Mathematical Society.
16. Dawson, D.A., Hochberg, K.J. and Vinogradov, V. (1995a) On path properties of super-2 processes II. In M.G. Cranston and M. Pinsky (eds), Proceedings of Symposia in Pure and Applied Mathematics, Series 57, pp. 385-403. Providence, Rt: American Mathematical Society.
17. Dawson, D.A., Hochberg, K.J. and Vinogradov, V. (1995b) High-density limits of hierarchically structured branching-diffusing populations. Preprint.
18. Dynkin, B. (1991a) Branching particle systems and superprocesses. Ann. Probab., 19, 1157-1194.
19. Dynkin, B. (1991b) Path processes and historical super-processes. Probab. Theory Related Fields, 90, 89-115.
20. Etheridge, A. (1993) Limiting behaviour of two-level measure branching. Adv. Appl. Probab., 25, 773-782.
21. Fernandez, B. and Gorostiza, L.G. (1990) Hydrodynamic and fluctuation limits of branching particle systems with changes of mass. Bol. Soc. Mat. Mexičana, 35, 25-37.
22. Fernandez, B. and Gorostiza, L.G. (1992) Convergence of generalized semimartingales to a continuous process. In G. Da Prato and L. Tubaro (eds), Stochastic Partial Differential Equations and Applications, pp. 158-164. Pitman Res. Notes Math. Ser. 268. Harlow: Longman.
23. Gorostiza, L. G. (1983) High density limit theorems for infinite systems of unsealed branching Brownian motions. Ann. Probab., 11, 374-392.
24. Gorostiza, L.G. (1995) Fluctuations of a two-level critical branching system. In A. Etheridge (ed.), Stochastic Partial Differential Equations, Edinburgh 1994. London Math. Soc. Lecture Note Ser. 216, pp. 120-125. Cambridge: Cambridge University Press.
25. Gorostiza, L.G., Hochberg, K.J. and Wakolbinger, A. (1995) Persistence ofa critical super-2 process. J. Appl. Probab., 32, 534-540.
26. Gorostiza, L.G. and López-Mimbela, J.A. (1990) The multitype measure branching process. Adv. Appl. Probab., 22, 49-67.
27. Gorostiza, L.G. and López Mimbela, J.A. (1993) A convergence criterion for measure-valued processes, and applications to continuous superprocesses. In D. Nualart and M. Sanz Sole (eds), Barcelona Seminar on Stochastic Analysis, 1992, pp. 62-72. Progr. Probab. 32. Boston: Birkhäuser.
28. Gorostiza, L.G. and Roelly, S. (1991) Some properties of the multitype measure branching process. Stochastic Process. Appl., 37, 259-274.
29. Gorostiza, L.G., Roelly, S. and Wakolbinger, A. (1992) Persistence of critical multitype particle and measure branching processes, Probab. Theory Related Fields, 92, 313-335.
30. Gorostiza, L.G. and Wakolbinger, A. (1991) Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab., 19, 266-288.
31. Gorostiza, L.G. and Wakolbinger, A. (1994) Long time behavior of critical branching particle systems and applications. In D.A. Dawson (ed.), Measure-Valued Process, Stochastic Partial Differential Equations and interacting Systems, pp. 119-137. CRM Proc. Lecture Notes 5. Providence, RI: American Mathematical Society.
32. Greven, A. (1991) A phase transition for the coupled branching process I. Probab. Theory Related Fields, 87, 417-458.
33. Henry, D. (1981) Geometric Theory of Semilmear Parabolic Equations. Lecture Notes in Math. 840. New York: Springer-Verlag.
34. Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L. (1993) White Noise. An Infinite Dimensional Calculus. Dordrecht: Kluwer.
35. Hochberg, K.J. (1995) Hierarchically structured branching populations with spatial motion. Rocky Mountain J. Math., 25, 269-283.
36. Kochberg, K.J. and Wakolbinger, A. (1995) Non-persistence of two-leve! branching particle systems in low dimensions. In A. Etheridge (ed.), Stochastic Partial Differential Equations, Edinburgh 1994. London Math. Soc. Lecture Note Ser. 216, pp. 126-140. Cambridge: Cambridge University Press.
37. Holley, R. and Stroock, D.W. (1978) Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motion. Publ. Res. Inst. Math. Sci., Kyoto Univ., 14, 741-788.
38. Iscoe, I. (1986) A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields, 71,85-116.
39. Kallenberg, O. (1983) Random Measures. London: Academic Press.
40. Le Gall, J.F. (1991) A class of path-valued Markov processes and its application to superprocesses. Probab. Theory Related Fields, 95, 25-46.
41. López-Mimbela, J.A. (1992) Fluctuation limits of multitype branching random fields. J. Multivariate Anal, 40, 56-83.
42. Mitoma, I. (1983a) On the sample continuity of S'-process. J. Math. Soc. Japan, 35, 629-636.
43. Mitoma, I. (1983b) Tightness of probabilities in C(0, lļ,
44. Perkins, E. (1995) On the Martingale Problem for Interactive Measure-Valued Branching Diffusions. Mem. Amer. Math. Soc. 549. Providence, RI: American Mathematical Society.
45. Stöckl, A. and Wakolbinger, A, (1994) On clan recurrence and transience in time stationary branching Brownian particle systems. In D.A, Dawson (ed.), Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems. CRM Proc. Lecture Notes 5, pp. 213-219. Providence, RI: American Mathematical Society.
46. Treves, F. (1967) Topological Vector Spaces, Distributions and Kernels. New York: Academic Press.
47. Wu, Y. (1991) Dynamic particle systems and multilevel measure branching processes. Ph.D. thesis, Carleton University, Ottawa.
48. Wu, Y. (1993) Multilevel birth and death particle system and its continuous diffusion. Adv. Appl. Probab., 25, 549-569.
49. Wu, Y. (1994a) Asymptotic behavior of two level measure branching processes. Ann. Probab., 22, 854-874.
50. Wu, Y. (1994b) A three level particle system and existence of general multilevel measure-valued processes. In D.A. Dawson (ed.), Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems. CRM Proc. Lecture Notes 5, pp. 233-241. Providence, RI: American Mathematical Society