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Dynamical systems defined on point processes

Part of: Stochastic processes Measure-theoretic ergodic theory

Published online by Cambridge University Press:  14 July 2016

Laurence A. Baxter
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Abstract

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

Keywords

Poisson processautocovariance functioncharacterizationflow of probability density functionssemidynamical systemPerron-Frobenius operatorlinear Boltzmann equationchaos

MSC classification

Primary: 60G55: Point processes
Secondary: 28D20: Entropy and other invariants
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 581 - 588
Copyright
Copyright © Applied Probability Trust 1998 

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References

Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. New York, Springer.Google Scholar
Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. New York, John Wiley.Google Scholar
Lasota, A., and Mackey, M. C. (1994). Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd edn. New York, Springer.CrossRefGoogle Scholar
Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press, Cambridge.Google Scholar
Provatas, N., and Mackey, M. C. (1991a). Asymptotic periodicity and banded chaos. Physica D, 53, 295–318.Google Scholar
Provatas, N., and Mackey, M. C. (1991b). Noise-induced asymptotic periodicity in a piecewise linear map. J. Statist. Phys. 63, 585–612.CrossRefGoogle Scholar