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% Author Package file for use with AMS-LaTeX 1.2
%
%\documentstyle{amsart}
\controldates{24-OCT-1997,24-OCT-1997,24-OCT-1997,24-OCT-1997}
 
\documentclass{era-l}
\issueinfo{3}{17}{January}{1997}
\dateposted{October 28, 1997}
\pagespan{110}{113}
\PII{S 1079-6762(97)00033-4}
\def\copyrightyear{1997}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
 
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
 
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\begin{document}
 
\title[A deterministic displacement theorem
for Poisson processes]{A deterministic displacement theorem \\
for Poisson processes}
 
\author{Oliver Knill}
\address{Department of Mathematics, University of Arizona, Tucson, AZ 85721}
\curraddr{Department of Mathematics, University of Texas, Austin, TX 78712}
%\email{knill@ma.utexas.edu}
\email{knill@math.utexas.edu}

\subjclass{Primary 58F05, 82C22, 60G55; Secondary 70H05, 60K35, 60J60}
\date{July 28, 1997}
\keywords{Hamiltonian dynamics, Vlasov dynamics, Poisson point process}

\commby{Mark Freidlin}

%\issueinfo{3}{1}{October}{1997}

\copyrightinfo{1997}{American Mathematical Society}

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

\section{Poisson processes}
Some families of distributions in probability theory
are natural because they appear in central limit theorems, because they 
maximize entropy and because they are invariant
under convolutions. An example is the Gaussian 
distribution on $\mathbb R$: adding and then normalizing independent 
random variables of the same distribution give a new distribution 
with larger relative entropy except at the attractive Gaussian fixed point 
of this renormalization map.  A similar fixed point on 
discrete distributions is the Poisson distribution on $\mathbb N$. 
While the Gaussian distribution gives rise to stochastic
processes like Brownian motion, the Poisson distribution occurs
in random point processes like the Poisson process which is
defined as follows: let $S$ be some 
Euclidean space and let $P$ be a finite measure on $S$. A Poisson 
process consists of a collection of random finite sets $\Pi(\omega)$ 
for which the random variables 
%$N_B(\omega) =  P[S] \cdot |\Pi(\omega) \cap B|/|\Pi(\omega)|$ counting the 
$N_B(\omega) =  |\Pi(\omega) \cap B|$ counting the
relative number of points in a Borel set $B \subset S$ are Poisson 
distributed with mean $P[B]$ and such that $N_{B_j}$
are independent for disjoint sets $B_j \subset S$. The mean measure $P$
determines the process because a typical 
point set $\Pi(\omega)$ is obtained by picking randomly a natural number 
$d=d(\omega)$ with probability $e^{-P[S]} (d!)^{-1}$, 
choosing then independently $d$ random points 
$a_1(\omega), \dots, a_d(\omega)$ from $S$ with law $P$
and forming the set 
%$\Pi(\omega)=\{a_1(\omega), \dots, a_d(\omega) \}$. 
$\Pi(\omega)=\{a_1(\omega), \dots, a_d(\omega) \}$. We call $P(\omega)$
the counting measure on the finite set $\Pi(\omega)$. It satisfies
$P(\omega)[B]=N_B(\omega)$.

Poisson processes occur frequently in applications, for example as models for
traffic on freeways, stars in galaxies or populations of plants. 
What happens if the point sets are evolved in time possibly with 
interaction? 
To have the displaced process also as a Poisson process is
useful: the mean measure $P^t$ at a later time determines how 
a typical point configuration $\Pi^t(\omega)$ looks like. 
With a law for the evolution of $P^t$, one can 
disregard the possibly complicated microscopic motion of 
$\Pi^t(\omega)$ and still have the information which is needed at any 
time.  For example, a typical star distribution of a galaxy at a later time 
could be determined from $P^t$ without integrating the Newton 
equations of the individual stars. 

\section{The Hamiltonian displacement theorem}
A known stability of the Poisson process is when
each particle is dislocated independently of the other particles. 
This is the displacement theorem of Bartlett, formulated in more generality in 
\cite{Kingman}. An example is when the distribution of the 
displaced position of a point is determined by a Markov transition probability 
density function $\rho^t(x,y)=\rho^t(x-y)$.  The mean measure of the displaced
Poisson process satisfies then $P^t = P \star \rho^t$. 

The independence of the dislocations cannot be weakened in all generality as is
indicated in \cite{Kingman}. The next theorem assures that the Poisson property is 
robust under deterministic Hamiltonian evolutions with 
nice smooth potentials $V$ for which one has global existence of 
the dynamics. 

\begin{theorem}[The Hamiltonian displacement theorem] 
Given a smooth finite \linebreak
 measure $P$ of compact support in the phase space
$S=T^*M$ of $M=\mathbb R^q$. It defines a random Poisson process. 
Assume the random finite point sets 
$\Pi(\omega)=\{ (x_1,y_1), \dots, (x_d,y_d) \} \subset S$
are evolved with a smooth deterministic Hamiltonian d-body dynamics
%$$
\[ \dot{x}_j=\dot{y}_j, \dot{y}_j = - d^{-1} \sum_{k \neq j} \nabla V(x_j-x_k)  \; . 
%$$
\]
Then the displaced process 
does not produce correlation forces.
\end{theorem} 

The probability space  
of the Poisson process is the big
phase space $\Omega = \bigcup_{d=0}^{\infty} S^d$ with Borel 
$\sigma$-algebra and probability measure 
$Q= \sum_{d=0}^{\infty} e^{-P[S]} (d!)^{-1} P_d$, where 
$P_d = P \times \cdots \times P$ is the product measure of the 
multi-particle space $S^d$. For $\omega \in S^d \subset \Omega$ one gets 
$\Pi(\omega)=\{ \omega_1, \dots, \omega_d \}$. 

The Hamiltonian displacement theorem
extends to more general cases. Particles can have different 
masses, interact with different type of potentials, move on 
more general manifolds with possible boundaries, have time-dependent potentials, 
evolve with more general Hamiltonians, and 
move with a nearest neighbor interaction;
they can additionally be exposed to external fields or interact with 
suitable $k$-body interactions. One can also add dissipation or evolve with a
relativistic particle interaction.

\section{The evolution of the mean measure} 
The Vlasov equation or collisionless Boltzmann equation 
%$$
\[ \dot{P}^t + y \cdot \nabla_x P^t
    - E(x,P^t) \cdot \nabla_y P^t = 0 
%$$
\]
with $E(x,P^t)=\int_S \nabla V(x-x') \; dP^t(x',y')$ 
is a fundamental evolution equation in stellar dynamics or plasma
physics (see \cite{Spohn},
\cite{DSS}). It has also applications in nuclear 
physics or in models of supersonic flows. 
It is a nonlinear partial differential equation. While it
is usually used to evolve continuous measures, it can describe 
the evolution $P^t$ of any measure on the phase 
space $S$. For example, if $\tilde{P}^t(\omega)$ is the counting measure
on a finite point set $\Pi^t(\omega)$ which evolves with the 
Hamiltonian $d$-body evolution with potential $V$, then $\tilde{P}^t(\omega)$
solves weakly the Vlasov equation. We use the tilde 
in order to distinguish the measure 
$\tilde{P}^t(\omega)$ coming from the Hamiltonian evolution and the random 
measures $P^t(\omega)$ defined by the Poisson process.

\begin{theorem}
The mean measure $P^t$ of the displaced process satisfies a
nonlinear Vlasov type equation
\begin{equation}
  \label{meanvlasov}
  \dot{P}^t + y \cdot \nabla_x P^t - \nabla_y \cdot E(P^t) = 0   , 
\end{equation}
where $E(P^t) = \int_{\Omega} E(x,P^t(\omega)) P^t(\omega) \; dQ(\omega)$
 with 
\[
E(x,P^t(\omega))
=P^t(\omega)[S]^{-1} \int_S \nabla V(x-x') \,
dP^t(\omega)
\]
and where $P^t(\omega)$ are the random measures defined by $P^t$. 
The measure $E(P^t)$ has a smooth density and is defined by $P^t$ alone. 
\end{theorem}

The nonlinear evolution equation~(\ref{meanvlasov})
encodes the statistics of the finite dimensional particle evolutions 
$\Pi^t(\omega)$ similar as diffusion equations encode stochastic
particle motions. The mean Vlasov equation is an average of Vlasov 
equations for the discrete measures $\tilde{P}^t(\omega)$. It approaches 
the Vlasov equation for the smooth measure in the limit when the particle
density converges to infinity. 
The mean measure $P^t$ would also satisfy the actual Vlasov equation 
if each particle configuration $\Pi(t)$ would move under the averaged field 
$E(x,P)$ of the mean measure $P$. In this case, the displacement
result is an easy consequence of the mapping theorem in the theory of Poisson 
processes \cite{Kingman}: if $X^t$ is the symplectic transformations on $S$ satisfying
the characteristic equations so that 
$X^t_* P^0 = P^t$ solves the Vlasov equation, then the process at time 
$t$ would be the image of the process at time $t=0$ under the map $X^t$. 

\section{Interacting Brownian particles} 
The Hamiltonian displacement theorem is related to the already mentioned 
displacement theorem of Bartlett which is the same statement under the assumption that
each particle is displaced independently of the other particles without
interaction. For example, 
if each particle moves along an independent Brownian path 
then the displaced process stays a Poisson process and 
the mean measure $P^t$ satisfies the heat equation 
$\dot{P} = c \Delta P$.  

The Hamiltonian displacement theorem can be combined with 
Bartlett's theorem and applied to Hamiltonian $d$-body dynamics 
driven by white noise.
The Hamilton equations are now replaced by the stochastic differential
equations
%$$
\[ dx_j = y_j dt, \qquad
   dy_j = - \frac{1}{d} \sum_{k \neq j} \nabla V(x_j-x_k) dt + c dB_j  ,
%$$
\]
where $\{B_j\}_{j=1}^d$ is a collection of independent Brownian motions and 
$c$ is a real parameter. 

\begin{theorem}
Assume the point configurations $\Pi(\omega)$ of a Poisson process evolve
as interacting Brownian particles; then the displaced process
has a correlation measure which satisfies a McKean-Vlasov
diffusion equation 
%$$
\[ \dot{P} + y \cdot \nabla_x P-\nabla_y \cdot E(P)- \frac{c^2}{2} \Delta P=0 
%$$
\]
which becomes the mean Vlasov equation in the limit $c=0$ when the particles evolve 
as a Hamiltonian system. 
\end{theorem} 

Adding to the Hamiltonian evolution the dynamics of interacting Brownian particles 
provides according to Nelson's stochastic mechanics a quantization of
classical mechanics. Mathematically, 
the limit $c \to 0$ could
serve as a technical regularization tool to investigate the mean 
Vlasov equation. As in the case $c=0$, one obtains in the limit of an infinite 
particle density the McKean-Vlasov equation
%$$
\[ \dot{P} + y \cdot \nabla_x P
    - E(x,P) \cdot \nabla_y P -  \frac{c^2}{2} \Delta P=0  
%$$
\]
which is for $c=0$ the Vlasov equation. 

\section{To the proof of the Hamiltonian displacement theorem} 
In this section we say something about the proof of
the Hamiltonian displacement theorem. 
 
R\'enyi's theorem in the theory of Poisson processes \cite{Kingman}
states that if $Q[N_B=~0]$ $ = \exp(-P[B])$ for all
finite unions of cubes $B$, then
$N_B$ is defined by a Poisson process. This result, obtained
by R\'enyi in 1967, simplifies
the task to verify the conditions for the Poisson process. Especially, it
frees us from the duty to check the independence property. 
 
Let $X^t(\omega): S \rightarrow S$
be the one-parameter family of symplectomorphisms
on $S$ defined by $X^t(\omega)=(f^t(\omega),g^t(\omega))$
which satisfy the Hamilton equations
%$$
\[  \dot{f}^t(\omega,x,y) = g^t(\omega, x,y),
    \qquad\dot{g}^t(\omega,x,y) =  - E(f^t(\omega),P^0(\omega))(x,y) \; . 
%$$
\]
These equations
are the characteristic equations of the Vlasov dynamics and the push-forwards
$\tilde{P}^t(\omega)=X^t_*(\omega) P^0(\omega)$ solve weakly
the Vlasov equation. 
 
Consider a family of Poisson processes with mean measure $P^t$, where $P^t$
satisfies the mean Vlasov equation.
To prove the theorem, we compare the random point process
%$$
\[ X^t(\omega) \Pi^0(\omega)  \; 
%$$
\]
with the Poisson point process $\Pi^t(\omega)$. The main task is to show
%$$
\[ \frac{d}{dt} Q[ N_{X^{-t}(\omega)(B)}(\omega) = 0 ] =
   \frac{d}{dt} \exp(-P^t[B])       \;  
%$$
\]
for any finite union of cubes $B \subset S$. With this and with
R\'enyi's result the displacement theorem follows.
%In this section we outline shortly the idea of the proof of 
%the Hamiltonian displacement theorem. 
%
%R\'enyi's theorem in the theory of Poisson processes \cite{Kingman}
%states that if $Q[N_B=0] = \exp(-P[B])$ for all 
%finite unions of cubes $B$, then 
%$N_B$ is defined by a Poisson process. This result, obtained
%by R\'enyi in 1967, simplifies 
%the task to verify the conditions for the Poisson process. Especially, it
%frees us from the duty to check the independence property. 
%
%Let $X^t(\omega): S \rightarrow S$ 
%be the one-parameter family of symplectomorphisms
%on $S$ defined by $X^t(\omega)=(f^t(\omega),g^t(\omega))$ 
%which satisfy the Hamilton equations
%%$$\dot{f}^t(\omega,x,y) = g^t(\omega, x,y), 
%    \dot{g}^t(\omega,x,y) =  - E(f^t(\omega),P^0(\omega))(x,y) \; , 
%$$
%where 
%%$$E(f^t(\omega),P^0(\omega))(x,y)=
%  \int_S \nabla V(f(\omega,x,y)-f(\omega,x',y')) \; dP^0(\omega,x',y')  \; . 
%$$
%These equations
%are the characteristic equations of the Vlasov dynamics and the push-forwards 
%$\tilde{P}^t(\omega)=X^t_*(\omega) P^0(\omega)$ solve weakly 
%the Vlasov equation. 
%Let $X^t=(f^t,g^t)$ solve the Hamilton equations 
%$\dot{X}^t=\int_{\Omega} \dot{X}^t(\omega) \; dQ^t(\omega)$ obtained by 
%averaging the vector fields of $\dot{X}^t(\omega)$. 
%(It will follow from the theorem that 
%$P^t= \int_{\Omega} P^t(\omega) \; dQ^t(\omega)
%    = \int_{\Omega} \tilde{P}^t(\omega) \; dQ^t(\omega)$ for all $t$ and
%that $X^t_* P^0$ solves the mean Vlasov equation.) 
%
%By the mapping theorem in the theory of Poisson processes, 
%$N_{X^t B}$ is defined by a Poisson process with mean measure $X^t_* P^0$. 
%To prove the theorem, we compare the random point process 
%%$$X^t(\omega) \Pi^0(\omega)  \; 
%$$
%with the Poisson point process $X^t \Pi^0(\omega)$ and show that
%$$ \frac{d}{dt} Q^t[ N_{X^t(\omega)(B)}(\omega) = 0 ] = 
%   \frac{d}{dt} Q^t[ N_{X^t(B)}(\omega) = 0 ]   \;  $$
%for any finite union of cubes $B \subset S$. With this and with
%R\'enyi's result the displacement theorem is proven.  
%
%There are two $\omega$-dependencies in the random variable
%$ \omega \mapsto N_{X^t(\omega)(B)}(\omega)$. The
%Poisson property at time $t=0$ means that the displacement
%$X^t(\omega)(B)$ of the set $B$ is essentially independent of the
%random variable
%$\omega \mapsto 1_{ \{ \eta \; | \; N_B(\eta) =  0 \} }(\omega) \}$. 
%Most of the work goes in making this separation rigorous. One can  
%take then the conditional expectation of
%$$ \omega=(\omega_1,\omega_2)
%            \mapsto \frac{d}{dt} Q^t[ N_{X^t(\omega)(B)}(\omega) = 0 ]
%             = \frac{d}{dt} Q^t[ N_{X^t(\omega_1)(B)}(\omega_2) = 0 ] $$
%to obtain $\omega \mapsto \frac{d}{dt} Q^t[ N_{X^t(B)}(\omega) = 0 ]$. 

\bibliographystyle{amsplain}
\begin{thebibliography}{1}

\bibitem{DSS}
L. A. Bunimovich et al.,
%\newblock
{\em Dynamical systems {II}}, 
%\newblock
 Encyclopaedia of Mathematical Sciences,
  vol. 2 (Ya. G. Sinai, ed.),  Springer-Verlag, Berlin, 1989.
 \MR{91i:58079} 
\bibitem{Kingman}
J. F. C. Kingman,
%\newblock
 {\em Poisson processes}, Oxford Studies in
  Probability, vol. 3,
%\newblock
 Clarendon Press, Oxford University Press, New York, 1993.
\MR{94a:60052}
\bibitem{Spohn}
H. Spohn
%\newblock
 {\em Large scale dynamics of interacting particles},
%\newblock
 Texts and monographs in physics, Springer-Verlag, New York, 1991.

\end{thebibliography}

\end{document}
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