A deterministic displacement theorem for Poisson processes
Author:
Oliver Knill
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 110-113
MSC (1991):
Primary 58F05, 82C22, 60G55; Secondary 70H05, 60K35, 60J60
DOI:
https://doi.org/10.1090/S1079-6762-97-00033-4
Published electronically:
October 28, 1997
MathSciNet review:
1475535
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: 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 $\dot {P} + y \cdot \nabla _x P-\nabla _y \cdot E(P)- c \Delta P=0$.
- 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. MR 1024068
- J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207584
- H. Spohn Large scale dynamics of interacting particles, Texts and monographs in physics, Springer-Verlag, New York, 1991.
- L. A. Bunimovich et al., Dynamical systems II, Encyclopaedia of Mathematical Sciences, vol. 2 (Ya. G. Sinai, ed.), Springer-Verlag, Berlin, 1989.
- J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, Clarendon Press, Oxford University Press, New York, 1993.
- H. Spohn Large scale dynamics of interacting particles, Texts and monographs in physics, Springer-Verlag, New York, 1991.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
58F05,
82C22,
60G55,
70H05,
60K35,
60J60
Retrieve articles in all journals
with MSC (1991):
58F05,
82C22,
60G55,
70H05,
60K35,
60J60
Additional Information
Oliver Knill
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721
Address at time of publication:
Department of Mathematics, University of Texas, Austin, TX 78712
Email:
knill@math.utexas.edu
Keywords:
Hamiltonian dynamics,
Vlasov dynamics,
Poisson point process
Received by editor(s):
July 28, 1997
Published electronically:
October 28, 1997
Communicated by:
Mark Freidlin
Article copyright:
© Copyright 1997
American Mathematical Society