Abstract
Stochastic dynamics associated with Gibbs measures on an infinite product of compact Riemannian manifolds is constructed. The probabilistic representations for the corresponding Feller semigroups are obtained. The uniqueness of the dynamics is proved.
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References
S. Albeverio, Yu. Kondratiev, and M. Röckner, “Dirichlet operators via stochastic analysis,” J. Funct. Anal., 128, No. 1, 102–138, (1995).
S. Albeverio, Yu. Kondratiev, and M. Röckner, “Quantum fields, Markov fields and stochastic quantization,” in: Stochastic Analysis: Mathematics and Physics, NATO ASI, Academic Press, New York (1995), pp. 3–29.
D. Stroock and B. Zegarlinski, “The equivalence of the logarithmic Sobolev inequality and Dobrushin-Shlosman mixing condition,” Commun. Math. Phys., 144, 303–323 (1992).
D. Stroock and B. Zegarlinski, “The logarithmic Sobolev inequality for continuous spin systems on a lattice,” J. Funct. Anal., 104, 299–326 (1992).
S. Albeverio, Yu. Kondratiev, and M. Röckner, “Uniqueness of the stochastic dynamics for continuous spin systems on a lattice,” J. Funct. Anal., 133, No. 1, 10–20 (1995).
A. Val. Antonjuk and A. Vic. Antonjuk, “Smoothness properties of semigroups for Dirichlet operators of Gibbs measures,” J. Funct. Anal., 127, No. 2, 390–430 (1995).
S. Albeverio, A. Val. Antonjuk, A. Vic. Antonjuk, and Yu. Kondratiev, “Stochastic dynamics in some lattice spin systems,” Methods of Functional Analysis and Topology, 1, No. 1, 3–28 (1995).
S. Albeverio, A. Daletskii, and Yu. Kondratiev, “A stochastic differential equation approach to some lattice models on compact Lie groups,” Random Operators and Stochastic Equations, 4, No. 3, 227–237 (1996).
K. D. Elworthy, “Geometric aspects of diffusions on manifolds,” Lect. Notes Math., 1362, 276–425 (1988).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland-Kodansha, Amsterdam (1989).
G. Leha and G. Ritter, “On solutions of stochastic differential equations with discontinuous drift in Hilbert space,” Math. Ann., 270, 109–123.
P. R. Halmos, A Hilbert Space Problem Book, Springer Verlag, Berlin (1982).
Yu. Dalecky and S. Fomin, Measures and Differential Equations in Infinite Dimensional Space, Kluwer, Dordrecht (1991).
R. Holley and D. Stroock, “Diffusions on the infinite dimensional torus,” J. Funct. Anal., 42, 29–63 (1981).
Yu. M. Berezansky and Yu. G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Kluwer, Dordrecht (1992).
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Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 326–337, March, 1997.
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Albeverio, S., Daletskii, A.Y. & Kondrat’ev, Y.G. Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds. Ukr Math J 49, 360–372 (1997). https://doi.org/10.1007/BF02487239
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DOI: https://doi.org/10.1007/BF02487239