Markov evolutions and hierarchical equations in the continuum. I: One-component systems

Journal of Evolution Equations

Volumn 9, Issue 2, 2009, Pages 197-233

  Finkelshtein, Dmitri L ^a   Kondratiev, Yuri G ^b   Oliveira, Maria João ^c,d  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

^b BIELEFELD UNIVERSITY   (Germany)

^c UNIVERSIDADE ABERTA   (Portugal)

^d UNIVERSITY OF LISBON   (Portugal)

Author keywords

Birth and death process; Configuration spaces; Contact model; Continuous system; Generating functional; Gibbs measure; Glauber dynamics; Hopping particles; Kawasaki dynamics; Markov generator; Markov process; Stochastic dynamics; Voter model

Indexed keywords

EID: 67349226154     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-009-0007-9     Document Type: Article

References (33)

1. Yu.M. Berezansky Yu.G. Kondratiev T. Kuna E.W. Lytvynov 1999 On a spectral representation for correlation measures in configuration space analysis Methods Funct. Anal. Topology 5 4 87 100
2. N. N. Bogoliubov. Problems of a Dynamical Theory in Statistical Physics. Gostekhisdat, Moscow, 1946. (in Russian). English translation in J. de Boer and G. E. Uhlenbeck (editors), Studies in Statistical Mechanics, vol. 1, pp. 1-118. North-Holland, Amsterdam, 1962.
3. C. Cocozza M. Roussignol 1979 Unicitè d'un processus de naissance et mort sur la droite réelle Ann. Inst. H. Poincaré Sect. B 15 93 106
4. D. L. Finkelshtein, Yu. G. Kondratiev, and O. Kutovyi. Individual based model with competition in spatial ecology. SIAM J. Math. Anal. (to appear).
5. D.L. Finkelshtein Yu.G. Kondratiev E.W. Lytvynov 2007 Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics Random Oper. Stochastic Equations 15 105 126
6. D. L. Finkelshtein, Yu. G. Kondratiev, and M. J. Oliveira. Markov evolutions and hierarchical equations in the continuum II. Multicomponent systems. In preparation, 2009.
7. E. Glötzl 1981 Time reversible and Gibbsian point processes. I. Markovian spatial birth and death processes on a general phase space Math. Nachr. 102 217 222
8. E. Glötzl 1982 Time reversible and Gibbsian point processes. II. Markovian particle jump processes on a general phase space Math. Nachr. 106 62 71
9. R. Holley D.W. Stroock 1978 Nearest neighbor birth and death processes on the real line Acta Math. 140 103 154
10. Yu.G. Kondratiev T. Kuna 2002 Harmonic analysis on configuration space I. General theory Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 2 201 233
11. Yu.G. Kondratiev T. Kuna 2003 Correlation functionals for Gibbs measures and Ruelle bounds Methods Funct. Anal. Topology 9 1 9 58
12. Yu. G. Kondratiev and T. Kuna. Harmonic analysis on configuration space II. Bogoliubov functional and equilibrium states. In preparation, 2009.
13. Yu.G. Kondratiev T. Kuna M.J. Oliveira 2004 On the relations between Poissonian white noise analysis and harmonic analysis on configuration spaces J. Funct. Anal. 213 1 1 30 (Pubitemid 38881887)
14. Yu.G. Kondratiev T. Kuna M.J. Oliveira 2006 Holomorphic Bogoliubov functionals for interacting particle systems in continuum J. Funct. Anal. 238 2 375 404 (Pubitemid 44089130)
15. Yu. G. Kondratiev, T. Kuna, M. J. Oliveira, J. L. Silva, and L. Streit. Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems. Preprint, 2008. http://www.math.uni-bielefeld.de/sfb701/preprints/ sfb08082.pdf.
16. Yu.G. Kondratiev O. Kutovyi R.A. Minlos 2008 On non-equilibrium stochastic dynamics for interacting particle systems in continuum J. Funct. Anal. 255 1 200 227
17. Yu.G. Kondratiev O. Kutoviy S. Pirogov 2008 Correlation functions and invariant measures in continuous contact model Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 2 231 258
18. Yu. Kondratiev O. Kutoviy E. Zhizhina 2006 Nonequilibrium Glauber-type dynamics in continuum J. Math. Phys. 47 11 113501
19. Yu. Kondratiev E. Lytvynov 2005 Glauber dynamics of continuous particle systems Ann. Inst. H. Poincaré Probab. Statist. 41 685 702 (Pubitemid 40721624)
20. Yu.G. Kondratiev E. Lytvynov M. Röckner 2007 Equilibrium Kawasaki dynamics of continuous particle systems Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 2 185 209
21. Yu. G. Kondratiev, E. Lytvynov, and M. Röckner. Non-equilibrium stochastic dynamics in continuum: the free case. Condensed Matter Physics (to appear).
22. Yu. Kondratiev A. Skorokhod 2006 On contact processes in continuum Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 2 187 198 (Pubitemid 43876764)
23. T. Kuna. Studies in Configuration Space Analysis and Applications. PhD thesis, Bonner Mathematische Schriften Nr.324, University of Bonn, 1999.
24. A. Lenard 1973 Correlation functions and the uniqueness of the state in classical statistical mechanics Commun. Math. Phys. 30 35 44
25. A. Lenard 1975 States of classical statistical mechanical systems of infinitely many particles I Arch. Rational Mech. Anal. 59 219 239
26. A. Lenard 1975 States of classical statistical mechanical systems of infinitely many particles II Arch. Rational Mech. Anal. 59 241 256
27. Liggett T.M.: Interacting Particle Systems. Springer Verlag, New York (1985)
28. Liggett T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer Verlag, Berlin (1999)
29. H.W. Lotwick B.W. Silverman 1981 Convergence of spatial birth-and-death processes Math. Proc. Cambridge Philos. Soc. 90 1 155 165
30. J. Møller 1989 On the rate of convergence of spatial birth-and-death processes Ann. Inst. Statist. Math. 41 565 581
31. X.X. Nguyen H. Zessin 1979 Integral and differential characterizations of the Gibbs process Math. Nachr. 88 105 115
32. M. J. Oliveira. Configuration Space Analysis and Poissonian White Noise Analysis. PhD thesis, Faculty of Sciences, University of Lisbon, 2002.
33. C. Preston. Spatial birth-and-death processes. In Proceedings of the 40th Session of the International Statistical Institute (Warsaw, 1975), volume 2, 1975. Bull. Inst. Internat. Statist. 46 (1975), 371-391.