## Electronic Journal of Probability

### Multiple Space-Time Scale Analysis For Interacting Branching Models

#### Abstract

We study a class of systems of countably many linearly interacting diffusions whose components take values in $[0, \inf)$ and which in particular includes the case of interacting (via migration) systems of Feller's continuous state branching diffusions. The components are labelled by a hierarchical group. The longterm behaviour of this system is analysed by considering space-time renormalised systems in a combination of slow and fast time scales and in the limit as an interaction parameter goes to infinity. This leads to a new perspective on the large scale behaviour (in space and time) of critical branching systems in both the persistent and non-persistent cases and including that of the associated historical process. Furthermore we obtain an example for a rigorous renormalization analysis.

#### Article information

Source
Electron. J. Probab., Volume 1 (1996), paper no. 14, 84 pp.

Dates
Accepted: 28 February 1996
First available in Project Euclid: 25 January 2016

https://projecteuclid.org/euclid.ejp/1453756477

Digital Object Identifier
doi:10.1214/EJP.v1-14

Mathematical Reviews number (MathSciNet)
MR1423467

Zentralblatt MATH identifier
0890.60093

Rights

#### Citation

Dawson, Donald; Greven, Andreas. Multiple Space-Time Scale Analysis For Interacting Branching Models. Electron. J. Probab. 1 (1996), paper no. 14, 84 pp. doi:10.1214/EJP.v1-14. https://projecteuclid.org/euclid.ejp/1453756477

#### References

• J. Baillon, P. Clement, A. Greven, F. den Hollander, 1. On the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions: The compact case, Canadian Journal of Mathematics 47, (1995), 3-27. 2. The noncompact case.
• J. T. Cox, K. Fleischmann, A. Greven, Comparison of interacting diffusions and an application to their ergodic theory, To appear Probab. Theory Rel. Fields, (1996).
• J. T. Cox, A. Greven, T. Shiga, Finite and infinite systems of interacting diffusions, Probab. Theory Rel. Fields, (1995).
• D. A. Dawson, Measure-valued Markov Processes, In: Ecole d'Ete de Probabilites de Saint Flour XXI, Lecture Notes in Mathematics 1541, (1993), 1-261, Springer-Verlag.
• D. A. Dawson and A. Greven, Multiple time scale analysis of hierarchically interacting systems, In: A Festschrift to honor G. Kallianpur, (1993), 41-50, Springer-Verlag.
• D. A. Dawson and A. Greven, Multiple time scale analysis of interacting diffusions, Probab. Theory Rel. Fields 95, (1993), 467-508.
• D. A. Dawson and A. Greven, Hierarchical models of interacting diffusions: multiple time scale phenomena. Phase transition and pattern of cluster-formation, Probab. Theory Rel. Fields, 96, (1993), 435-473.
• D. A. Dawson and P. March, Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems, J. Funct. Anal. 132, (1995), 417-472.
• D. A. Dawson, A. Greven, J. Vaillancourt, Equilibria and Quasiequilibria for Infinite Collections of Interacting Fleming-Viot processes. Transactions of the American Math. Society, volume 347, no. 7, (1995), 2277-2360.
• D. A. Dawson and E. A. Perkins, Historical Processes, Memoirs of the A.M.S., 454, (1991).
• P. Donnelly and T. G. Kurtz, A countable representation of the Fleming-Viot measure-valued diffusion, preprint, (1991).
• J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, (1984).
• R. Durrett, An infinite particle system with additive interactions, Adv. Appl. Probab. 11, (1979), 355-383.
• R. Durrett, Ten Lectures on Particle Systems, In: Ecole d'Ete de Probabilites de Saint Flour XXIII, Lecture Notes in Mathematics, Springer-Verlag, (1993).
• R. Durrett and C. Neuhauser, Particle systems and reaction diffusion equations, Ann. Probab., to appear, (1993).
• W. Feller, An Introduction to Probability Theory and its Applications Vol. II, Wiley & Sons, (1992).
• J. Fleischmann, Limiting distributions for branching random fields, Trans. Amer. Math. Soc. 239, (1978), 353-389.
• K. Fleischmann and A. Greven, Diffusive clustering in an infinite system of hierarchically interacting diffusions, Probab. Theory Rel. Fields, 98, (1994), 517-566.
• G. Gauthier, Multilevel systems of bilinear stochastic differential equations, Preprint: Technical Report Series of the Laboratory for Research in Statistics and Probability, No. 254, (1994).
• L. G. Gorostiza and A. Wakolbinger, Convergence to equilibrium of critical branching particle systems and superprocesses and related nonlinear partial differential equations, Acta Appl. Math. 27, (1992), 269-291.
• P. Jagers, Branching Processes with Biological Applications, J. Wiley, (1975).