## The Annals of Probability

### Different clustering regimes in systems of hierarchically interacting diffusions

Achim Klenke

#### Abstract

We study a system of interacting diffusions

indexed by the hierarchical group $\Xi$, as a genealogical two genotype model [where $x _\xi(t)$ denotes the frequency of, say, type A] with hierarchically determined degrees of relationship between colonies. In the case of short interaction range it is known that the system clusters. That is, locally one genotype dies out. We focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel $a(\dot,\dot)$ via its recurrent potential kernel. One of these regimes will be further investigated by mean-field methods. For general interaction range we shall also relate the behavior of large finite systems, indexed by finite subsets $\Xi_n$ of, $\Xi$ to that of the infinite system. On the way we will establish relations between hitting times of random walks and their potentials.

#### Article information

Source
Ann. Probab., Volume 24, Number 2 (1996), 660-697.

Dates
First available in Project Euclid: 11 December 2002

https://projecteuclid.org/euclid.aop/1039639358

Digital Object Identifier
doi:10.1214/aop/1039639358

Mathematical Reviews number (MathSciNet)
MR1404524

Zentralblatt MATH identifier
0862.60096

#### Citation

Klenke, Achim. Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. 24 (1996), no. 2, 660--697. doi:10.1214/aop/1039639358. https://projecteuclid.org/euclid.aop/1039639358

#### References

• Arratia, R. (1982). Coalescing Brownian motion and the voter model on Z. Unpublished manuscript.
• Baillon, J.-B., Cl´ement, Ph., Greven, A. and den Hollander, F. (1995). On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions. Part I: the compact case. Canad. J. Math. 47 3-27.
• Bramson, M. and Griffeath, D. (1980). Clustering and dispersion rates for some interacting particle sy stems on Z. Ann. Probab. 8 183-213.
• Cox, J. T. (1989). Coalescing random walks and voter model consensus times on the torus in Zd. Ann. Probab. 17 1333-1366.
• Cox, J. T. and Durrett, R. (1995). Hy brid zones and voter model interfaces. Bernoulli 1 343-370.
• Cox, J. T., Fleischmann, K. and Greven, A. (1996). Comparison of interacting diffusions and applications to their ergodic theory. Probab. Theory Related Fields. To appear.
• Cox, J. T. and Greven, A. (1990). On the long term behavior of some finite particle sy stems. Probab. Theory Related Fields 85 195-237.
• Cox, J. T. and Greven, A. (1991). On the long term behavior of finite particle sy stems: a critical dimension example. In Random Walks, Brownian Motion and Interacting Particle Sy stems: A Festschrift in Honor of Frank Spitzer (R. Durrett and H. Kesten, eds.). Birkh¨auser, Boston. Cox, J. T. and Greven, A. (1994a). Ergodic theorems for infinite sy stems of locally interacting diffusions. Ann. Probab. 22 833-853. Cox, J. T. and Greven, A. (1994b). The finite sy stems scheme: an abstract theorem and a new example. CRM Proceedings and Lecture Notes 5 55-67.
• Cox, J. T., Greven, A. and Shiga, T. (1994). Finite and infinite sy stems of interacting diffusions. Probab. Theory Related Fields. To appear.
• Cox, J. T. and Griffeath, D. (1986). Diffusive clustering in the two dimensional voter model. Ann. Probab. 14 347-370.
• Dawson, D. A. and G¨artner, J. (1988). Long time behavior of interacting diffusions. In Stochastic Calculus in Application: Proceedings of the Cambridge Sy mposium, 1987 (J. R. Norris, ed.). 29-54, Longman, Harlow, UK. Dawson, D. A. and Greven, A. (1993a). Multiple Time Scale Analy sis of Hierarchically Interacting Sy stems. A Festschrift to Honor G. Kallianpur 41-50. Springer, New York. Dawson, D. A. and Greven, A. (1993b). Multiple time scale analysis of interacting diffusions. Probab. Theory Related Fields 95 467-508.
• Dawson, D. A., Greven, A. and Vaillancourt, J. (1995). Equilibria and quasiequilibria for infinite sy stems of interacting Fleming-Viot processes. Trans. Amer. Math. Soc. 347 2277- 2361.
• Durrett, R. and Schonmann, R. H. (1988). The contact process on a finite set, II. Ann. Probab. 16 1570-1583.
• Erd ¨os, P. and Tay lor, S. J. (1960). Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11 137-162.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.
• Fleischmann, K. and Greven, A. (1994). Diffusive clustering in an infinite sy stem of hierarchically interacting diffusions. Probab. Theory Related Fields 98 517-566.
• Kemeny, J. G., Snell, J. S. and Knapp, A. W. (1976). Denumerable Markov chains, 2nd ed. Springer, New York.
• Liggett, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
• Sawy er, S. (1976). Results for the stepping stone model for migration in population genetics. Ann. Probab. 4 699-728.
• Shiga, T. (1980). An interacting sy stem in population genetics. J. Math. Ky oto Univ. 20 213-242.
• Shiga, T. and Shimizu, A. (1980). Infinite dimensional stochastic differential equations and their applications. J. Math. Ky oto Univ. 20 395-416.
• Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.
• Tribe, R. (1993). The long term behavior of a stochastic PDE. Preprint 74, Berlin.
• Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Ky oto Univ. 11 155-167.