Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

Forum Mathematicum

Volumn 18, Issue 1, 2006, Pages 9-43

  Kondratiev, Yuri ^a,b   Lytvynov, Eugene ^c   Röckner, Michael ^a,b  

^a BIELEFELD UNIVERSITY   (Germany)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

^c SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646726814     PISSN: 09337741     EISSN: 09337741     Source Type: Journal    
DOI: 10.1515/FORUM.2006.002     Document Type: Article

References (47)

1. Albeverio S., Daletskii A., Lytvynov E.: Laplace operators on differential forms over configuration spaces. J. Geom. Phys. 37 (2001), 15-46
2. Albeverio S., Kondratiev Yu. G., Röckner M.: Differential geometry of Poisson spaces. C.R. Acad. Sci. Paris 323 (1996), 1129-1134
3. Alberverio S., Kondratiev Yu. G., Röckner M.; Canonical Dirichlet operator and distorted Brownian motion on Poisson spaces. C.R. Acad. Sci. Paris 323 (1996), 1179-1184
4. Alberverio S., Kondratiev Yu. G., Röckner M.: Analysis and geometry on configuration spaces. J. Funct. Anal. 154 (1998), 444-500
5. Alberverio S., Kondratiev Yu. G., Röckner M.: Analysis and geometry on configuration spaces. The Gibbsian case. J. Funct. Anal. 157 (1998), 242-291
6. Albeverio S., Röckner M.: Classical Dirichlet forms on topological vector spaces - Closability and a Cameron-Martin formula. J. Funct. Anal. 88 (1990), 395-436
7. Albeverio S., Röckner M.; Dirichlet form methods for uniqueness of martingale problems and applications. In: Stochastic Analysis. Proceedings of Symposia in Pure Mathematics (ed. by M. C. Cranston and M. A. Pinsky) 57, pp. 513-528. Am. Math. Soc., Providence, RI 1995
8. Berezansky Yu. M., Kondratiev Yu. G.: Spectral Methods in Infinite Dimensional Analysis. Kluwer Acad. Publ. Dordrecht-Boston-London 1994
9. Brox T.: Gibbsgleichgewichtsfluktuationen für einige Potentiallimiten. Ph.D. thesis, University of Heidelberg 1980
10. da Silva J. L., Kondratiev Yu. G., Röckner M.: On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems. Math. Nachr. 222 (2001), 141-157
11. Dynkin E. B.: Markov Processes. Springer-Verlag, Berlin 1965
12. Ethier S., Kurtz T.: Markov Processes. Wiley and Sons, New York 1986
13. Fritz J.: Gradient dynamics of infinite point systems. Ann. Prob. 15 (1987), 478-514
14. Fukushima M.: Dirichlet Forms and Symmetric Markov Processes. North-Holland, Amsterdam 1980
15. Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. W. de Gruyter, Berlin-New York 1994
16. Georgii H. O.: Canonical Gibbs Measures. Lecture Notes in Math. 760. Springer-Verlag, Berlin-New York 1979
17. Grothaus M., Kondratiev Yu. G., Lytvynov E., Röckner M.: Scaling limit of stochastic dynamics in classical continuous systems. Ann. Prob. 31 (2003), 1494-1532
18. Guo M. Z., Papanicolaou G.: Bulk diffusion for interacting Brownian particles. In: Statistical Physics and Dynamical Systems (Köszeg, 1984), pp. 41-48. Birkhäuser, Boston 1985
19. Kallenberg O.: Random Measures. Academic Press, San Diego 1975
20. Kondratiev Yu. G., Kuna T.: Harmonic analysis on configuration spaces I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 201-233
21. Kondratiev Yu. G., Kuna T., Oliveira M. J.: Non-equilibrium stochastic dynamics of continuous systems and Bogoliubov functionals. In preparation
22. Kondratiev Yu. G., Lytvynov E.: Glauber dynamics of continuous particle systems. Ann. Inst. H. Poincaré Prob. Statist., to appear
23. Kuna T.: Studies in Configuration Space Analysis and Applications. Ph.D. thesis, Bonn University 1999
24. Lang R.: Unendlich-dimensionale Wienerprozesse mit Wechselwirkung I, II. Z. Wahrsch. Verw. Gebiete 38 (1977), 55-72,
25. Lang R.: Z. Wahrsch. Verw. Gebiete, 39 (1977), 277-299
26. Lyons T. J., Zhang W. A.: A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Colloque Paul Lévy sur les processes stochastique, Asterisque 157-158 (1988), 249-272
27. Lyons T. J., Zhang W. A.: Decomposition of Dirichlet processes and its applications. Ann. Prob. 22 (1994), 494-524
28. Ma Z.-M., Röckner M.: An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin 1992
29. Ma Z.-M., Röckner M.: Construction of diffusions on configuration spaces. Osaka J. Math. 37 (2000), 273-314
30. Malyshev V. A., Minlos R. A.: Gibbs Random Fields: Cluster Expansion. Kluwer Academic Publisher, Dordrecht-Boston-London 1991
31. Minlos R. A.: Limiting Gibbs distributions. Funktsional'nyj Analiz i Ego Prilozhenija 1 (1967), no. 2, 60-73
32. Nguyen X. X., Zessin H.: Integral and differentiable characterizations of the Gibbs process. Math. Nachr. 88 (1979), 105-115
33. Osada H.: Dirichlet form approach to infinite-dimensional Wiener processes with singular interaction. Comm Math. Phys. 176 (1996), 117-131
34. Osada H.: An invariance principle for Markov processes and Brownian particles with singular interaction. Ann. Inst. Henri Poinaré 34 (1998), 217-247
35. Osada H.: Positivity of the self-diffusion matrix of interacting Brownian particles with hard core. Prob. Theory Relat. Fields 112 (1998), 53-90
36. Parthasarthy K. R.: Probability Measures on Metric Spaces. Academic Press, New York 1967
37. Röckner M.: Stochastic analysis on configuration spaces: basic ideas and recent results. In: New Directions in Dirichlet Forms (ed. by J. Jost et al.), pp. 157-231. Studies in Advanced Mathematics 8. Amer. Math. Soc., Providence, RI; International Press, Cambridge, MA 1998
38. Röckner M., Schmuland B.: Quasi-regular Dirichlet forms: examples and counter-examples. Canad. J. Math. 47 (1995), 165-200
39. Röckner M., Schmuland B.: A support property for infinite-dimensional interacting diffusion processes. C. R. Acad. Sci. Paris 326 (1998), Série I, 359-364
40. Rost H.: Hydradynamik gekoppelter Diffusionen: Fluktuationen im Gleichgewicht. In: Dynamics and Processes (ed. by Ph. Blanchard and L. Streit). LNM 1031, pp. 97-107. Springer-Verlag, Berlin 1981
41. Ruelle D.: Statistical Mechanics. Rigorous Results. Benjamin, Amsterdam 1969
42. Ruelle D.: Superstable interaction in classical statistical mechanics. Comm. Math. Phys. 18 (1970), 127-159
43. Shiga T.: A remark on infinite-dimensional Wiener processes with interactions. Z. Wahrsch. Verw. Gebiete 47 (1979), 299-304
44. Schmuland B., Sun W.: The law of large numbers and the law of the iterated logarithm for infinite dimensional interacting processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 489-503
45. Spohn H.: Equilibrium fluctuations for interacting Brownian particles. Comm. Math. Phys. 103 (1986), 1-33
46. Vershik A. M., Gelfand I. M., Graev M. I.: Representations of the group of diffeomorphisms. Russian Math. Surveys 30 (1975), no. 6, 1-50
47. Yoshida M. W.: Construction of infinite-dimensional interacting diffusion processes through Dirichlet forms. Prob. Theory Related Fields 106 (1996), 265-297