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Stochastic differential equations in a scale of Hilbert spaces. (English) Zbl 1406.60087
Summary: A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in $${\mathbb{R} }^{n}$$.
##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 46E99 Linear function spaces and their duals
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##### References:
 [1] S. Albeverio, A. Daletskii, Yu. Kondratiev, Stochastic equations and Dirichlet operators on product manifolds. Infinite Dimensional Analysis, Quantum Probability and Related Topics6 (2003), 455-488. · Zbl 1049.60055 [2] S. Albeverio, A. Daletskii, Yu. Kondratiev, Stochastic analysis on product manifolds: Dirichlet operators on differential forms. J. Funct. Anal.176 (2000), no. 2, 280-316. · Zbl 0970.58020 [3] S. Albeverio, Yu. Kondratiev, M. Röckner, Analysis and geometry on configuration spaces: The Gibbsian case, J. Funct. Anal.157 (1998), 242–291. · Zbl 0931.58019 [4] R. Barostichi, A. Himonas, G. Petronilho, Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems, J. Funct. Anal.270 (2016), 330–358. · Zbl 1331.35299 [5] T. Bodineau, I. Gallagher, L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math.203 (2016), 493–553. · Zbl 1337.35107 [6] A. Bovier, Statistical Mechanics of Disordered Systems. A Mathematical Perspective (Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006). · Zbl 1108.82002 [7] D. Crisan, D. Holm, Wave breaking for the Stochastic Camassa-Holm equation, Physica D: Nonlinear Phenomena376-377 (2018), 138-143. · Zbl 1398.35305 [8] R. Dalang, M. Dozzi, F. Flandoli, F. Russo (eds.), Stochastic Analysis: A Series of Lectures, Centre Interfacultaire Bernoulli, January–June 2012, Ecole Polytechnique Fédérale de Lausanne, Switzerland, Progress in Probability (2015), Birkhauser. · Zbl 1330.60004 [9] A. Daletskii, D. Finkelshtein, Non-equilibrium particle dynamics with unbounded number of interacting neighbors, J. Stat. Phys. (2018), published on-line http://link.springer.com/article/10.1007/s10955-018-2159-x. · Zbl 1405.82020 [10] A. Daletskii, Yu. Kondratiev, Yu. Kozitsky, T. Pasurek, Gibbs states on random configurations, J. Math. Phys. 55 (2014), 083513. · Zbl 1301.82021 [11] A. Daletskii, Yu. Kondratiev, Yu. Kozitsky, T. Pasurek, Phase Transitions in a quenched amorphous ferromagnet, J. Stat. Phys. 156 (2014), 156-176. · Zbl 1298.82074 [12] G. Da Prato, J. Zabczyk, Stochastic Differential Equations in Infinite Dimensions, Cambridge 1992. · Zbl 0761.60052 [13] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series229, University Press, Cambridge, 1996. · Zbl 0849.60052 [14] K. Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics596, Springer 1977. · Zbl 0361.34050 [15] D. Finkelshtein, Around Ovsyannikov’s method, Methods of Functional Analysis and Topology21 (2015), No. 2, 134-150. · Zbl 1340.35177 [16] D. Finkelshtein, Yu. Kondratiev, O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal.262 (2012), 1274-1308. · Zbl 1250.47090 [17] J. Fritz, C. Liverani, S. Olla, Reversibility in Infinite Hamiltonian Systems with Conservative Noise, Commun. Math. Phys.189 (1997), 481 - 496. · Zbl 0893.70011 [18] J. Inglis, M. Neklyudov, B. Zegarliński, Ergodicity for infinite particle systems with locally conserved quantities, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012), No. 1, 1250005. · Zbl 1268.60120 [19] G. Kallianpur, I. Mitoma, R. L. Wolpert, Diffusion equations in duals of nuclear spaces, Stochastics and Stochastic Reports, 29 (1990), No. 2, 285-329. · Zbl 0702.60056 [20] G. Kallianpur, Jie Xiong, Stochastic differential equations in infinite dimensional spaces, Lecture notes-monograph series26, Institute of Mathematical Statistics 1995. [21] D. Klein and W. S. Yang, A characterization of first order phase transitions for superstable interactions in classical statistical mechanics, J. Stat. Phys.71 (1993), 1043-1062. · Zbl 0935.82519 [22] O. Lanford, Time evolution of large classical systems, Lecture notes in physics 38, pp. 1-111, Springer (1975) [23] O. Lanford, J. Lebowitz, E. Lieb, Time Evolution of Infinite Anharmonic Systems, J. Stat. Phys.16 (1977), No. 6, 453–461. [24] T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry12 (1977), 629-633. · Zbl 0368.35007 [25] S. Romano and V. A. Zagrebnov, Orientational ordering transition in a continuous-spin ferrofluid, Phys. A253 (1998), 483–497. [26] D. Ruelle, Superstable interactions in classical statistical mechanics, Commun. Math. Phys.18 (1970), 127–159. · Zbl 0198.31101 [27] M. V. Safonov, The Abstract Cauchy-Kovalevskaya Theorem in a Weighted Banach Space, Comm. Pure Appl. Math.XLVIII (1995), 629-637. · Zbl 0836.35004
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