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

DOI:
https://doi.org/10.1090/S0002-9947-1973-0326868-3

MathSciNet review:
0326868

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Abstract | References | Similar Articles | Additional Information

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, implies 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 related to *Q* is introduced, and we show that with the appropriate topology the map taking *Q* to 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0326868-3

Keywords:
Infinite particle system,
random counting measure,
tagged particle distribution,
point process

Article copyright:
© Copyright 1973
American Mathematical Society