Finite and infinite systems of interacting diffusions: Cluster formation and universality properties

Mathematische Nachrichten

Volumn 192, Issue , 1998, Pages 105-124

  Cox, John Ted ^a   Greven, Andreas ^c   Shiga, Tokuzo ^b  

^a SYRACUSE UNIVERSITY   (United States)

^b TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^c UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Clustering; Finite particle systems; Interacting diffusions

Indexed keywords

EID: 0040040163     PISSN: 0025584X     EISSN: None     Source Type: Journal    
DOI: 10.1002/mana.19981920107     Document Type: Article

References (29)

1. ALDOUS, D.: Stoppings Times and Tightness II, Ann. Probab. 17 (1989), 586-595
2. BAILLON, J.-B., CLÉMENT, PH., GREVEN, A., and DEN HOLLANDER, F.: On the Attracting Orbit of a Non - Linear Transformation Arising from Renormalization of Hierarchically Interacting Diffusions, Part I: The Compact Case Can. J. Math. 47 (1995), 3-27
3. BAILLON, J.-B., CLÉMENT, PH., GREVEN, A., and DEN HOLLANDER, F.: Part II: The Noncompact Case (1994), Journal of Functionl Analysis Vol. 146, No. 1, (1997)
4. d., Ann. Probab. 17 (1989), 1333-1366
5. COX., J. T., FLEISCHMANN, K., and GREVEN, A.: A Comparison of Interacting Diffusions and an Application to Their Ergodic Theory, Probab. Th. Rel. Fields 105 (1996), 513-528
6. COX, J. T., and GREVEN, A.: On the Long Term Behaviour of Some Finite Particle Systems, Probab. Th. Rel. Fields 85 (1990), 195-237
7. COX, J. T., and GREVEN, A.: On the Long Term Behaviour of Some Finite Particle Systems: A Critical Dimension Example, Random Walks, Brownian Motion, and Interacting Particle Systems, R. DURRETT and H. KESTEN eds., Birkhäuser, 1991
8. COX, J. T., and GREVEN, A.: Ergodic Theorems for Infinite Systems of Interacting Diffusions, Ann. Probab. Vol. 22, No. 3, (1994), 833-853
9. COX, J. T., and GREVEN, A.: The Finite Systems Scheme: An Abstract Theorem and a New Example, CRM Proceedings and Lecture Notes 5, D. DAWSON ed., (1994), 55-68, Measure - Valued Processes, Stochastic Partial Differential Equations and Interacting Systems.
10. COX, J. T., GREVEN, A. and SHIGA, T.: Finite and Infinite Systems of Interacting Diffusions. Part I, Prob. Th. Rel. Fields 102 (1995), 165-197
11. DAWSON, D. A., and GREVEN, A.: Multiple Time Scale Analysis of Interacting Diffusions, Prob. Th. Rel. Fields 95 (1993), 467-508
12. DAWSON, D. A., and GREVEN, A.: Models of Hierarchical Interacting Diffusions: Multiple Time Scale Phenomena, Phase Transition and Patterns of Cluster Formation, Prob. Th. Rel. Fields 96 (1993), 435-474
13. DAWSON, D. A., and GREVEN, A.: Multiple Space - Time Scale Analysis for Branching Systems (1995), to appear: Electronic Journal of Probability, Vol. 1 (1996), paper 14, 1-84
14. COX, J. T., and GRIFFEATH, D.: Diffusive Clustering in the Two Dimensional Voter Model, Ann. Probab. 14 (1986), 347-370
15. ETHIER, S. N., and KURTZ, T. G.: Markov Processes - Characterization and Convergence, John Wiley & Sons, 1986
16. FLEISCHMANN, K., and GREVEN, A.: Diffusive Clustering in an Infinite System of Hierarchically Interacting Diffusions, Prob. Th. Rel. Fields 98 (1994), 517-566
17. KLENKE, A.: Different Clustering Regimes in Systems of Hierarchically Interacting Diffusions, Ann. Prob. Vol. 24, No. 2, (1996), 660-697
18. KOPIETZ, A.: Clusterformation für wechselwirkende Diffusionen, Preprint, 1995
19. KONNO, N., and SHIGA, T.: Stochastic Partial Differential Equations for Some Measure-Valued Diffusions, Prob. Th. Rel. Fields (1988), 201-225
20. LIGGETT, T. M.: Interacting Particle Systems, Springer - Verlag, 1985
21. NOTOHARA, M., and SHIGA, T.: Convergence to Genetically Uniform State in Stepping Stone Models of Population Genetics, J. Math. Bio. 10 (1980), 281-294
22. SHIGA, T.: An Interacting System in Population Genetics I, J. Math. Kyoto Univ. 20 (1980), 213-243
23. SHIGA, T.: An Interacting System in Population Genetics II, J. Math. Kyoto Univ. 20 (1980), 723-733
24. SHIGA, T.: A Certain Class of Infinite - Dimensional Diffusion Processes Arising in Population Genetics, J. Math. Soc. Japan (1987), 17-25
25. SHIGA, T.: Ergodic Theorems and Exponential Decay of Sample Paths for Certain Interacting Diffusion Systems, Osaka J. Math. (1992), 789-807
26. SHIGA, T., and SHIMIZU, A.: Infinite - Dimensional Stochastic Differential Equations and Their Applications, J. Math. Kyoto Univ. 20 (1980), 395-416
27. SHIGA, T., and UCHIYAMA, K.: Stationary States and Their Stability of the Stepping Stone Model Involving Mutation and Selection, Prob. Th. Rel. Fields (1986), 87-117
28. SPITZER, F.: Principles of Random Walk, Springer - Verlag, 1976
29. WATANABE, S., and YAMADA, T.: On the Uniqueness of Solutions of Stochastic Differential Equations, J. Math. Kyoto Univ. (1971), 155-167