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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite particle systems

Authors: Sidney C. Port and Charles J. Stone
Journal: Trans. Amer. Math. Soc. 178 (1973), 307-340
MSC: Primary 60K35; Secondary 82.60
MathSciNet review: 0326868
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Abstract: We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected ``at random'' from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences, $ {P_n} \to P$ implies $ {Q_n} \to Q$ and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q' at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected ``at random'' from those initially in some set. We also show that Q' is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution $ {Q_0}$ related to Q is introduced, and we show that with the appropriate topology the map taking Q to $ {Q_0}$ is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.

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Keywords: Infinite particle system, random counting measure, tagged particle distribution, point process
Article copyright: © Copyright 1973 American Mathematical Society

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