Electronic Journal of Probability

Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution

Anita Winter

Full-text: Open access

Abstract

We consider two types of measure-valued branching processes on the lattice $Z^d$. These are on the one hand side a particle system, called branching random walk, and on the other hand its continuous mass analogue, a system of interacting diffusions also called super random walk. It is known that the long-term behavior differs sharply in low and high dimensions: if $d\leq 2$ one gets local extinction, while, for $d\geq 3$, the systems tend to a non-trivial equilibrium. Due to Kallenberg's criterion, local extinction goes along with clumping around a 'typical surviving particle.' This phenomenon is called clustering. A detailed description of the clusters has been given for the corresponding processes on $R^2$ in Klenke (1997). Klenke proved that with the right scaling the mean number of particles over certain blocks are asymptotically jointly distributed like marginals of a system of coupled Feller diffusions, called system of tree indexed Feller diffusions, provided that the initial intensity is appropriately increased to counteract the local extinction. The present paper takes different remedy against the local extinction allowing also for state-dependent branching mechanisms. Instead of increasing the initial intensity, the systems are described under the Palm distribution. It will turn out together with the results in Klenke (1997) that the change to the Palm measure and the multiple scale analysis commute, as $t\to\infty$. The method of proof is based on the fact that the tree indexed systems of the branching processes and of the diffusions in the limit are completely characterized by all their moments. We develop a machinery to describe the space-time moments of the superprocess effectively and explicitly.

Article information

Source
Electron. J. Probab., Volume 7 (2002), paper no. 13, 74 pp.

Dates
Accepted: 15 March 2002
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1463434886

Digital Object Identifier
doi:10.1214/EJP.v7-112

Mathematical Reviews number (MathSciNet)
MR1921742

Zentralblatt MATH identifier
1010.60077

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
infinite particle system superprocess interacting diffusion clustering Palm distribution grove indexed systems of diffusions grove indexed systems of branching models Kallenberg's backward tree

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Winter, Anita. Multiple Scale Analysis of Spatial Branching Processes under the Palm Distribution. Electron. J. Probab. 7 (2002), paper no. 13, 74 pp. doi:10.1214/EJP.v7-112. https://projecteuclid.org/euclid.ejp/1463434886


Export citation

References

  • Bhattacharya, R.N. and Rao, R.R. (1976), Normal approximation and asymptotic expansions, Wiley New-York.
  • Chauvin, B., Rouault, A. and Wakolbinger, A. (1991), Growing conditioned trees, Stoc. Proc. Appl., 39, 117-130.
  • Cox, J.T.; Fleischmann, K. and Greven, A. (1996), Comparison of interacting diffusions and application to their ergodic theory, Probab. Theor. Rel. Fields, 105, 513-528.
  • Cox, J.T. and Greven, A. (1994), Ergodic theorems for infinite systems of locally interacting diffusions, Ann. Probab., 22(2), 833-853.
  • Cox, J.T., Greven, A. and Shiga, T. (1995), Finite and infinite systems of interacting diffusions, Probab. Rel. Fields, 103, 165-197.
  • Cox, J.T. and Griffeath, D. (1986), Diffusive clustering in the two dimensional voter model, Ann. Probab., 14(2), 347-370.
  • Dawson, D.A. (1977), The critical measure diffusion process, Z. Wahrscheinlichleitstheorie verw. Gebiete, 40, 125-145.
  • Dawson, D.A. (1993), Measure-valued Markov processes, École d'Été de Probabilités de Saint Flour XXI – 1991, Lect. Notes in Math. 1541, 1-260, Springer-Verlag.
  • Dawson, D.A. and Greven, A. (1993), Multiple time scale analysis of hierarchical interacting systems, A Festschrift to honor G. Kallianpur, Springer New York, 41-50.
  • Dawson, D.A. and Greven, A. (1993), Multiple time scale analysis of interacting diffusions, Probab. Theory Relat. Fields 95, 467-508.
  • Dawson, D.A. and Greven, A. (1996), Multiple space-time analysis for interacting branching models, EJP 1, paper 14, 1-84.
  • Dawson, D.A., Greven, A. and Vaillancourt, J. (1995), Equilibria and quasi equilibria for infinite systems of interacting Fleming-Viot processes, Trans. AMS, 347(7), 2277-2361, Memoirs of the American Mathematical Society 93, 454.
  • Deuschel, J. D. (1988), Central limit theorem for an infinite lattice system of interacting diffusion processes, Ann. Probab. 16, 700-716.
  • Durrett, R. (1979), An infinite particle system with additive interactions, Adv. Appl. Prob., 11, 355-383.
  • Durrett, R. (1991), Probability: Theory and Examples, Duxbury Press, Belmont, California.
  • Dynkin, E.B. (1988), Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, Astérisque, 157-158, 147-171.
  • Etheridge, A. (1993), Asymptotic behavior of measure-valued critical branching processes, Proc. AMS, 118(4), 1251-1261.
  • Ethier, S.N. and Krone, S.M. (1995), Comparing Fleming-Viot and Dawson-Watanabe processes, Stoc. Proc. Appl., 60, 171-190.
  • Fleischman, J. (1978), Limiting distributions for branching random fields, Trans. AMS, 239, 353-389.
  • Fleischmann, K. and Greven, A. (1996), Time-space analysis of the cluster-formation in interacting diffusions, EJP 1, paper 6, 1-46.
  • Gorostiza, L.G., Roelly, S. and Wakolbinger, A. (1992), Persistence of critical multitype particle and measure branching processes, Probab. Theory Relat. Fields, 92, 313-335.
  • Gorostiza, L.G. and Wakolbinger, A. (1991), Persistence criteria for a class of critical branching particle systems in continuous time, Ann. Probab., 19(1), 266-288.
  • Holley, R. and Liggett, T.M. (1975), Ergodic theorems for weakly interacting systems and the voter model, Ann. Probab., 3, 643-663.
  • Hurwitz, A. (1964), Vorlesungen über allgemeine Funktionentheorie, Springer-Verlag.
  • Kallenberg, O. (1977), Stability of critical cluster fields, Math. Nachr., 77, 7-43.
  • Kallenberg, O. (1983), Random Measures, Akademie-Verlag Berlin.
  • Klenke, A. (1996) Different clustering regime in systems of hierarchically interacting diffusions,Ann. Probab., 24(2), 660-697. 
  • Klenke, A. (1997), Multiple scale analysis of clusters in spatial branching models, Ann. Probab., 25(4), 1670-1711.
  • Klenke, A. (1998) Clustering and invariant measures for spatial branching models with infinite variance, Ann. Probab., 26(3), 1057-1087.
  • Klenke, A. (2000), Diffusive clustering on interacting Brownian motions on $Z^2$, Stoch. Proc. Appl., 89(2), 261-268.
  • Kopietz, A. (1998), Diffusive Clusterformation für wechselwirkende Diffusionen mit beidseitig unbeschränktem Zustandsraum, Dissertation, Universität Erlangen-Nürnberg.
  • Lee, T. Y. (1991), Conditional limit distributions of critical branching Brownian motions, Ann. Probab., 19(1), 289-311.
  • Liggett, T.M. and Spitzer, F. (1981), Ergodic theorems for coupled random walks and other systems with locally interacting components, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 56, 443-468.
  • Matthes, K. (1972), Infinitely divisible point processes, in Stochastic point processes: Statistical Analysis, Theory, and Applications, Wiley Interscience, New-York/London/Sydney/Toronto, 384-404.
  • Roelly-Coppoletta, S. and Rouault, A. (1989), Processes de Dawson-Watanabe conditionné par le futur lointain, C.R.Acad.Sci.Paris, 309, Série I, 867-872.
  • Shiga, T. (1980), An interacting system in population genetics, Jour. Kyoto Univ., 20(2), 213-243.
  • Shiga, T. (1992), Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems, Osaka J. Math., 29, 789-807.
  • Sirjaev, A.N. (1988), Wahrscheinlichkeit, VEB Deutscher Verlag der Wissenschaften Berlin, 308.
  • Winter, A. (1999), Multiple scale analysis of spatial branching processes under the Palm distribution, Dissertation, Universität Erlangen-Nürnberg, URL: http://www.mi.uni-erlangen.de/~winter/.