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## A deterministic displacement theorem

for Poisson processes

### Oliver Knill

**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
$\dot{P} + y \cdot \nabla_x P-\nabla_y \cdot E(P)- c \Delta P=0$.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 110-113
- Publisher Identifier: S 1079-6762(97)00033-4
- 1991
*Mathematics Subject Classification*. Primary 58F05,
82C22, 60G55; Secondary 70H05, 60K35, 60J60
*Key words and phrases*. Hamiltonian dynamics, Vlasov
dynamics, Poisson point process
- Received by the editors July 28, 1997
- Posted on October 28, 1997
- Communicated by Mark Freidlin
- Comments (When Available)

**Oliver Knill**

Department of Mathematics, University of Arizona, Tucson, AZ 85721

*E-mail address:* `knill@math.utexas.edu`

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