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A deterministic displacement theorem for Poisson processes. (English) Zbl 0887.58018
Summary: We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an \(n\)-body evolution, the mean measure \(P\) satisfies a nonlinear Vlasov type equation \[ \dot{P} + y \cdot \nabla_x P - \nabla_y \cdot E(P) = 0. \] Combined with Bartlett’s theorem, the result generalizes to interacting Brownian particles, where the mean measure satisfies a McKean-Vlasov type diffusion equation \[ \dot{P} + y \cdot \nabla_x P-\nabla_y \cdot E(P)- c \Delta P=0. \]
MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
82C22 Interacting particle systems in time-dependent statistical mechanics
70H05 Hamilton’s equations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J60 Diffusion processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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References:
[1] L. A. Bunimovich, I. P. Cornfeld, R. L. Dobrushin, M. V. Jakobson, N. B. Maslova, Ya. B. Pesin, Ya. G. Sinaĭ, Yu. M. Sukhov, and A. M. Vershik, Dynamical systems. II, Encyclopaedia of Mathematical Sciences, vol. 2, Springer-Verlag, Berlin, 1989. Ergodic theory with applications to dynamical systems and statistical mechanics; Edited and with a preface by Sinaĭ; Translated from the Russian.
[2] J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. · Zbl 0771.60001
[3] H. Spohn, Large scale dynamics of interacting particles, Texts and monographs in physics, Springer-Verlag, New York, 1991. · Zbl 0742.76002
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