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Electronic Research Announcements
Electronic Research Announcements
ISSN 1079-6762

 

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
Published electronically: October 28, 1997
MathSciNet review: 1475535
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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 $\dot{P} + y \cdot \nabla _x P-\nabla _y \cdot E(P)- c \Delta P=0$.


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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

DOI: http://dx.doi.org/10.1090/S1079-6762-97-00033-4
PII: S 1079-6762(97)00033-4
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