Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T22:13:35.455Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for super-Brownian motion with immigration

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Mei Zhang
Mei Zhang*
Affiliation:
Beijing Normal University and Central University of Finance and Economics
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a large-deviation principle for super-Brownian motion with immigration, where the immigration is governed by the Lebesgue measure. We show that the speed function is t 1/2 for d = 1, t/logt for d = 2 and t for d ≥ 3, which is different from that of the occupation-time process counterpart (without immigration) and the model of random immigration.

Keywords

Large-deviation principlesuper-Brownian motionmeasure-valued processesimmigration processes

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 187 - 201
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arthreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Brezis, H., and Friedman, A. (1983). Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pure. Appl. 62, 73–97.Google Scholar
Dawson, D. A. (1977). The critical diffusion process. Z. Wahrscheinlichkeitsth. 40, 125–145.Google Scholar
Dawson, D. A. (1993). Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XX1 (Lecture Notes Math. 1542), Springer, Berlin, pp. 1–260.Google Scholar
Dawson, D. A., and Ivanoff, D. (1978). Branching diffusions and random measures. In Branching Processes (Adv. Prob. Relat. Topics 5), eds Joffe, A. and Ney, P., Marcel Dekker, New York, pp. 61–103.Google Scholar
Dembo, A., and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
Deuschel, J.-D., and Stroock, D. (1989). Large Deviations. Academic Press, Boston, MA.Google Scholar
Dynkin, E. B. (1989). Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc. 314, 255–282.Google Scholar
Gorostiza, L. G., and Li, Z. H. (2000). High density fluctuations of immigration branching particle systems. In Stochastic Models, Ottawa, 1998 (CMS Conf. Proc. 26), eds Gorostiza, L. G., and Ivanoff, B. G., American Mathematical Society, Providence, RI, pp. 159–171.Google Scholar
Hong, W.-M. (2003). Large deviation for the super-Brownian motion with super-Brownian immigration. J. Theoret. Prob. 16, 899–922.Google Scholar
Hong, W.-M., and Li, Z. H. (1999). A central limit theorem for super-Brownian motion with super-Brownian immigration. J. Appl. Prob. 36, 1218–1224.CrossRefGoogle Scholar
Hong, W.-M., and Li, Z. H. (2001). Fluctuations of a super-Brownian motion with randomly controlled immigration. Statist. Prob. Lett. 51, 285–291.Google Scholar
Iscoe, I. (1986). Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics 18, 197–243.Google Scholar
Iscoe, I., and Lee, T. Y. (1993). Large deviations for occupation times of measure-valued branching Brownian motions. Stoch. Stoch. Reports 45, 177–209.Google Scholar
Kamin, S., and Peletier, L. A. (1985). Singular solutions of the heat equation with absorption. Proc. Amer. Math. Soc. 95, 205–210.Google Scholar
Lee, T. Y. (1993). Some limit theorems for super-Brownian motion and semilinear differential equations. Ann. Prob. 21, 979–995.Google Scholar
Lee, T. Y., and Remillard, B. (1995). Large deviation for three dimensional super-Brownian motion. Ann. Prob. 23, 1755–1771.Google Scholar
Li, Z. H. (1996). Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Process. Appl. 62, 73–86.Google Scholar
Li, Z. H. (1998). Immigration processes associated with branching particle systems. Adv. Appl. Prob. 30, 657–675.Google Scholar
Li, Z. H., and Shiga, T. (1995). Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274.Google Scholar
Mytnik, L. (1998). Collision measure and collision local time for (α,d,β) superprocesses. J. Theoret. Probab. 11, 733–763.Google Scholar
Wang, Z. K. (1990). Power series expansions of superprocesses. Acta Math. Sci. (Chinese) 10, 361–364.Google Scholar
Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167.Google Scholar
Widder, D. V. (1941). The Laplace Transform. Princeton University Press.Google Scholar
Zhang, M. (2003). Moderate deviations for super-Brownian motion with immigration. To appear in Sci. China A.Google Scholar