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.
Of course, for any probability measure μ on \((\varGamma,\mathcal{B}(\varGamma ))\) one has B μ (0) = 1.
- 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.
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
Barroso, J.A.: Introduction to Holomorphy. Volume 106 of Mathematics Studies. North-Holland, Amsterdam (1985)
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)
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)
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)
Dineen, S.: Complex Analysis in Locally Convex Spaces. Volume 57 of Mathematics Studies. North-Holland, Amsterdam (1981)
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)
Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Individual based model with competition in spatial ecology. SIAM J. Math. Anal. 41, 297–317 (2009)
Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys. 141, 158–178 (2010)
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)
Finkelshtein, D.L., Kondratiev, Yu.G., Kutoviy, O.: Correlation functions evolution for the Glauber dynamics in continuum. Semigroup Forum 85, 289–306 (2012)
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)
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)
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)
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)
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)
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)
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)
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)
Kondratiev, Yu.G., Kuna, T., Oliveira, M.J.: Analytic aspects of Poissonian white noise analysis. Methods Funct. Anal. Topol. 8(4), 15–48 (2002)
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)
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)
Kondratiev, Yu., Kutoviy, O., Minlos, R.: On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J. Funct. Anal. 255, 200–227 (2008)
Kondratiev, Yu., Kutoviy, O., Zhizhina, E.: Nonequilibrium Glauber-type dynamics in continuum. J. Math. Phys. 47(11), 113501 (2006)
Kondratiev, Y., Skorokhod, A.: On contact processes in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2), 187–198 (2006)
Kuna, T.: Studies in configuration space analysis and applications. PhD thesis, Bonner Mathematische Schriften Nr. 324, University of Bonn (1999)
Lenard, A.: States of classical statistical mechanical systems of infinitely many particles II. Arch. Ration. Mech. Anal. 59, 241–256 (1975)
Treves, F.: Ovcyannikov Theorem and Hyperdifferential Operators. Volume 46 of Notas de Matemática. IMPA, Rio de Janeiro (1968)
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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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