Time-Space Analysis of the Cluster-Formation in Interacting Diffusions

Klaus Fleischmann (Weierstrass Institute for Applied Analysis and Stochastics)
Andreas Greven (Universitat Erlangen-Nurnberg)


A countable system of linearly interacting diffusions on the interval [0,1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).

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Pages: 1-46

Publication Date: April 8, 1996

DOI: 10.1214/EJP.v1-6


  • [CFG96] Cox, J.T., Fleischmann, K., Comparison of interacting diffusions and application to their ergodic theory. Probab. Theory Relat. Fields, to appear 1996. Math Review article not yet available.
  • [CG83] Cox, J.T., Griffeath, D., Occupation time limit theorem for the voter model. Ann. Probab., 11:876-893, 1983. Math. Review 85b:60096
  • [CG85] Cox, J.T., Griffeath, D., Occupation times for critical branching Brownian motions. Ann. Probab., 13(4):1108-1132, 1985. Math. Review 87h:60102
  • [CG86] Cox, J.T., Griffeath, D., Diffusive clustering in the two dimensional voter model. Ann. Probab., 14:347-370, 1986. Math. Review 87j:60146
  • [CG94] Cox, J.T., Greven, A., Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab., 22(2):833-853, 1994. Math. Review 95h:60158
  • [Cox89] Cox, J.T., Coalescing random walks and voter model consensus times on the torus in Z^d. Ann. Probab., 17(4):1333-1366, 1989. Math. Review 91d:60250
  • [Daw93] Dawson, D.A., Measure-valued Markov processes. In P.L. Hennequin, editor, École d'été de probabilités de Saint Flour XXI-1991, volume 1541 of Lecture Notes in Mathematics, pages 1-260, Berlin, Heidelberg, New York, 1993. Springer-Verlag. Math. Review 94m:60101
  • [DF88] Dawson, D.A., Fleischmann, K., Strong clumping of critical space-time branching models in subcritical dimensions. Stoch. Proc. Appl., 30:193-208, 1988. Math. Review 90d:60075
  • [DG93] Dawson, D.A., Greven, A., Hierarchical models of interacting diffusions: Multiple time scale phenomena, phase transitions and pattern of clusterformation. Probab. Theory Relat. Fields, 96:435-473, 1993. Math. Review 94k:60155
  • [DK93] Donelly, P., Kurtz, T.G., A countable representation of the Fleming-Viot measure-valued diffusion. Annals of Probability, to appear 1996. Math Review article not yet available.
  • [EF96] Evans, S.N., Fleischmann, K., Cluster formation in a stepping stone model with continuous, hierarchically structered sites. WIAS Berlin, Preprint Nr. 187, 1995, Ann. Probab., to appear 1996. Math Review article not yet available.
  • [EK86] Ethier, S.N., Kurtz, T.G. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. Math. Review 88a:60130
  • [FG94] Fleischmann, K. Greven, A., Diffusive clustering in an infinite system of hierarchically interacting diffusions. Probab. Theory Relat. Fields, 98:517-566, 1994. Math. Review 95j:60163
  • [Isc86] Iscoe, I., A weighted occupation time for a class of measure-valued critical branching Brownian motions. Probab. Theory Relat. Fields, 71:85-116, 1986. Math. Review 87c:60070
  • [Kin82] Kingman, J.F.C., The coalescent. Stoch. Proc. Appl., 13:235-248, 1982. Math. Review 84a:60079
  • [Kle95] Klenke, A., Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab., to appear 1995. Math Review article not yet available.
  • [Lig85] Liggett, T.M., Interacting Particle Systems. Springer-Verlag, New York, 1985. Math. Review 86e:60089
  • [Lin88] Lindqvist, H., Association of probability measures on partially ordered spaces. J. Multivariate Analysis, 26:111-132, 1988. Math. Review 89i:60008
  • [SF83] Sawyer, S., Felsenstein, J., Isolation by distance in a hierarchically clustered population. J. Appl. Probab., 20:1-10, 1983. Math. Review 84h:92022
  • [Shi80] Shiga, T., An interacting system in population genetics. J. Mat. Kyoto Univ., 20:213-242, 1980. Math. Review 82e:92029a
  • [SS80] Shiga, T., Shimizu, A., Infinite-dimensional stochastic differential equations and their applications. J. Mat. Kyoto Univ., 20:395-416, 1980. Math. Review 82i:60110
  • [Tav84] Tavaré, S., Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol., 26:119-164, 1984. Math. Review 86f:92017

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