We consider two types of convolutions (* and ★) of functions on spaces of finite configurations (finite subsets of a phase space) and study some of their properties. A relationship between the *-convolution and the convolution of measures on spaces of finite configurations is described. Properties of the operators of multiplication and derivation with respect to the *-convolution are investigated. We also present conditions under which the *-convolution is positive-definite with respect to the ★-convolution.
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References
N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946).
A. M. Vershik, I. M. Gel’fand, and M. I. Graev, “Representations of the group of diffeomorphisms,” Usp. Mat. Nauk, 30, No. 6(186), 3–50 (1975).
I. M. Gel’fand and G. E. Shilov, Generalized Functions. Properties and Operations [in Russian], Fizmatgiz, Moscow (1959); English translation: Academic Press, New York (1964).
J. W. Gibbs, Elementary Principles in Statistical Mechanics, Dover, New York (2002).
R. Dobrushin, Ya. Sinai, and Yu. Sukhov, “Dynamical systems in statistical mechanics,” in: VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends [in Russian], Vol. 2, VINITI, Moscow (1985), pp. 235–284.
J. L. Lebowitz and E. W. Montroll (editors), Nonequilibrium Phenomena. I. The Boltzmann Equation, North-Holland, Amsterdam (1983).
C. Cercignani, Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh (1975).
S. Albeverio, Y. Kondratiev, and M. Röckner, “Analysis and geometry on configuration spaces,” J. Funct. Anal., 154, No. 2, 444–500 (1998).
S. Albeverio, Y. Kondratiev, and M. Röckner, “Analysis and geometry on configuration spaces: the Gibbsian case,” J. Funct. Anal., 157, No. 1, 242–291 (1998).
A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer, Berlin (1991).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer, Berlin (1999).
Y. Kondratiev and T. Kuna, “Harmonic analysis on configuration space. I. General theory,” Infinite Dimension. Anal., Quantum Probab. Relat. Top., 5, No. 2, 201–233 (2002).
Y. Kondratiev, T. Kuna, and M. J. Oliveira, “Holomorphic Bogoliubov functionals for interacting particle systems in continuum,” J. Funct. Anal., 238, No. 2, 375–404 (2006).
T. Kuna, Studies in Configuration Space Analysis and Applications, Dissertation, Bonn (1999).
T. M. Liggett, Interacting Particle Systems, Springer, New York (1985).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, Berlin (1999).
E. Presutti, Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics, Springer, Berlin (2009).
M. Röckner, Stochastic Analysis on Configuration Spaces: Basic Ideas and Recent Results, American Mathematical Society, Providence (1998).
D. Ruelle, “Cluster property of the correlation functions of classical gases,” Rev. Modern Phys., 36, 580–584 (1964).
C. Y. Shen, “A functional calculus approach to the Ursell–Mayer functions,” J. Math. Phys., 13, 754–759 (1972).
C. Y. Shen, “On a certain class of transformations in statistical mechanics,” J. Math. Phys., 14, 1202–1204 (1973).
C. Y. Shen and D. S. Carter, “Representations of states of infinite systems in statistical mechanics,” J. Math. Phys., 12, 1263–1269 (1971).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 11, pp. 1547–1567, November, 2012.
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Finkel’shtein, D.L. On convolutions on configuration spaces. I. Spaces of finite configurations. Ukr Math J 64, 1752–1775 (2013). https://doi.org/10.1007/s11253-013-0749-y
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DOI: https://doi.org/10.1007/s11253-013-0749-y