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Particle picture approach to the self-intersection local time of density processes in

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  • Bojdecki, Tomasz
  • Talarczyk, Anna

Abstract

For a Poisson high-density system of independent motions in we consider the corresponding density process as the limit of fluctuations or, equivalently, the limit of the "total charge" if each particle is equipped with a random charge. We prove that under fairly general assumptions on the motions and on the intensity measure of the system, the self-intersection local time (SILT) of the density process can be expressed by means of intersections of pairs of evolving particles. This result helps to understand the interpretation and meaning of SILT. As an example, we discuss the cases of symmetric [alpha]-stable motions and fractional Brownian motions in detail.

Suggested Citation

  • Bojdecki, Tomasz & Talarczyk, Anna, 2005. "Particle picture approach to the self-intersection local time of density processes in," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 449-479, March.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:3:p:449-479
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    References listed on IDEAS

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    1. Talarczyk, Anna, 2001. "Self-intersection local time of order k for Gaussian processes in," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 17-72, November.
    2. Adler, Robert J. & Epstein, R., 1987. "Some central limit theorems for Markov paths and some properties of Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 24(2), pages 157-202, May.
    3. Adler, Robert J. & Lewin, Marica, 1992. "Local time and Tanaka formulae for super Brownian and super stable processes," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 45-67, May.
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    Cited by:

    1. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2015. "From intersection local time to the Rosenblatt process," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1227-1249, September.
    2. C. A. Fonseca-Mora, 2018. "Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space," Journal of Theoretical Probability, Springer, vol. 31(2), pages 867-894, June.

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