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A Survey on Bogoliubov Generating Functionals for Interacting Particle Systems in the Continuum

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From Particle Systems to Partial Differential Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 75))

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Abstract

This work is a survey on Bogoliubov generating functionals and their applications to the study of stochastic evolutions on states of continuous infinite particle systems.

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Notes

  1. 1.

    Of course, for any probability measure μ on \((\varGamma,\mathcal{B}(\varGamma ))\) one has B μ (0) = 1.

  2. 2.

    That is, \(G \upharpoonright _{\varGamma _{0}\setminus \varGamma _{\varLambda }}\equiv 0\), Γ Λ : = {η ∈ Γ: η ⊂ Λ}, for some bounded Borel set \(\varLambda \subseteq {\mathbb{R}}^{d}\) and there are C 1, C 2 > 0 such that \(\vert G(\eta )\vert \leq C_{1}{e}^{C_{2}\vert \eta \vert }\) for all η ∈ Γ 0.

  3. 3.

    That is, \(G \upharpoonright _{\varGamma _{ 0}\setminus \left (\bigsqcup _{n=0}^{N}\varGamma _{\varLambda }^{(n)}\right )} \equiv 0\), \(\varGamma _{\varLambda }^{(n)}:= \{\eta \in \varGamma:\eta \subset \varLambda \}{\cap \varGamma }^{(n)}\), for some \(N \in \mathbb{N}_{0}\) and for some bounded Borel set \(\varLambda \subseteq {\mathbb{R}}^{d}\).

References

  1. Barroso, J.A.: Introduction to Holomorphy. Volume 106 of Mathematics Studies. North-Holland, Amsterdam (1985)

    Google Scholar 

  2. Berns, C., Kondratiev, Yu.G., Kozitsky, Yu., Kutoviy, O.: Kawasaki dynamics in continuum: micro- and mesoscopic descriptions. J. Dyn. Diff. Equat. 25, 1027–1056 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berns, C., Kondratiev, Yu.G., Kutoviy, O.: Construction of a state evolution for Kawasaki dynamics in continuum. Anal. Math. Phys. 3(2), 97–117 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bogoliubov, N.N.: Problems of a Dynamical Theory in Statistical Physics. Gostekhisdat, Moscow (1946) (in Russian). English translation in de Boer, J., Uhlenbeck, G.E. (eds.) Studies in Statistical Mechanics, vol. 1, pp. 1–118. North-Holland, Amsterdam (1962)

    Google Scholar 

  5. Dineen, S.: Complex Analysis in Locally Convex Spaces. Volume 57 of Mathematics Studies. North-Holland, Amsterdam (1981)

    Google Scholar 

  6. Finkelshtein, D.L., Kondratiev, Yu.G., Kozitsky, Yu.: Glauber dynamics in continuum: a constructive approach to evolution of states. Discret. Contin. Dyn. Syst. 33(4), 1431–1450 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Individual based model with competition in spatial ecology. SIAM J. Math. Anal. 41, 297–317 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys. 141, 158–178 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Vlasov scaling for the Glauber dynamics in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(4), 537–569 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Correlation functions evolution for the Glauber dynamics in continuum. Semigroup Forum 85, 289–306 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Semigroup approach to non-equilibrium birth-and-death stochastic dynamics in continuum. J. Funct. Anal. 262, 1274–1308 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Finkelshtein, D., Kondratiev, Yu., Kutoviy, O., Zhizhina, E.: An approximative approach to construction of the Glauber dynamics in continuum. Math. Nachr. 285, 223–235 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Finkelshtein, D.L., Kondratiev, Yu.G., Oliveira, M.J.: Markov evolutions and hierarchical equations in the continuum. I: one-component systems. J. Evol. Equ. 9(2), 197–233 (2009)

    MATH  MathSciNet  Google Scholar 

  14. Finkelshtein, D.L., Kondratiev, Yu.G., Oliveira, M.J.: Glauber dynamics in the continuum via generating functionals evolution. Complex Anal. Oper. Theory 6(4), 923–945 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Finkelshtein, D.L., Kondratiev, Yu.G., Oliveira, M.J.: Kawasaki dynamics in the continuum via generating functionals evolution. Methods Funct. Anal. Topol. 18(1), 55–67 (2012)

    MATH  MathSciNet  Google Scholar 

  16. Garcia, N.L., Kurtz, T.G.: Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat. 1, 281–303 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Kondratiev, Yu., Kozitsky, Yu., Shoikhet, D.: Dynamical systems on sets of holomorphic functions. In: Complex Analysis and Dynamical Systems IV, Part 1. Contemporary Mathematics, vol. 553. American Mathematical Society, Providence (2011)

    Google Scholar 

  18. Kondratiev, Yu.G., Kuna, T.: Harmonic analysis on configuration space I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 201–233 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kondratiev, Yu.G., Kuna, T., Oliveira, M.J.: Analytic aspects of Poissonian white noise analysis. Methods Funct. Anal. Topol. 8(4), 15–48 (2002)

    MATH  MathSciNet  Google Scholar 

  20. Kondratiev, Yu.G., Kuna, T., Oliveira, M.J.: On the relations between Poissonian white noise analysis and harmonic analysis on configuration spaces. J. Funct. Anal. 213(1), 1–30 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kondratiev, Yu.G., Kuna, T., Oliveira, M.J.: Holomorphic Bogoliubov functionals for interacting particle systems in continuum. J. Funct. Anal. 238(2), 375–404 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kondratiev, Yu., Kutoviy, O., Minlos, R.: On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J. Funct. Anal. 255, 200–227 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kondratiev, Yu., Kutoviy, O., Zhizhina, E.: Nonequilibrium Glauber-type dynamics in continuum. J. Math. Phys. 47(11), 113501 (2006)

    Article  MathSciNet  Google Scholar 

  24. Kondratiev, Y., Skorokhod, A.: On contact processes in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2), 187–198 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kuna, T.: Studies in configuration space analysis and applications. PhD thesis, Bonner Mathematische Schriften Nr. 324, University of Bonn (1999)

    Google Scholar 

  26. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles II. Arch. Ration. Mech. Anal. 59, 241–256 (1975)

    MathSciNet  Google Scholar 

  27. Treves, F.: Ovcyannikov Theorem and Hyperdifferential Operators. Volume 46 of Notas de Matemática. IMPA, Rio de Janeiro (1968)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

    Dmitri L. Finkelshtein

  2. Universidade Aberta and CMAF, University of Lisbon, P 1649-003, Lisbon, Portugal

    Maria João Oliveira

Authors
  1. Dmitri L. Finkelshtein
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  2. Maria João Oliveira
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Correspondence to Maria João Oliveira .

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Editors and Affiliations

  1. UMR CNRS-UNS 7351, Université de Nice Sophia Antipolis, Laboratoire J.A.Dieudonné, Nice, France

    Cédric Bernardin

  2. Universidade do Minho, Centre of Mathematics, Braga, Portugal

    Patricia Gonçalves

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Finkelshtein, D.L., Oliveira, M.J. (2014). A Survey on Bogoliubov Generating Functionals for Interacting Particle Systems in the Continuum. In: Bernardin, C., Gonçalves, P. (eds) From Particle Systems to Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54271-8_6

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