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Random point fields with Markovian refinements and the geometry of fractally disordered media

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Abstract

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1.

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References

  1. J. Zaiman,Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems, Cambridge Univ. Press, Cambridge (1979).

    Google Scholar 

  2. I. I. Gikhman and A. V. Skorohod,Introduction to the Theory of Random Processes [in Russian] (2nd ed.), Nauka, Moscow (1977); English transl. prev. ed., Saunders, Philadelphia, Pa. (1969).

    MATH  Google Scholar 

  3. V. Z. Belen'kii,Geometrically Probabilistic Models of Crystallization [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  4. K. Matthes, J. Kerstan and J. Mecke,Infinitely Divisible Point Processes, Wiley, New York (1978).

    MATH  Google Scholar 

  5. H. Federer,Geometric Measure Theory, Springer, New York (1969).

    MATH  Google Scholar 

  6. P. R. Massopust,Fractal Functions, Fractal Surfaces, and Wavelets, Acad. Press, New York (1994).

    MATH  Google Scholar 

  7. M. F. Barnsley,Fractals Everywhere, Acad. Press, Orlando, Florida (1988).

    MATH  Google Scholar 

  8. K. R. Parthasarathy,Introduction to Probability and Measure, Macmillan Co. of India, New Delhi (1980).

    Google Scholar 

  9. A. Ya. Virchenko and A. Ya. Dulfan,Functional Materials,5, 471–474 (1998).

    Google Scholar 

  10. B. A. Sevast'yanov,Ramified Processes [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  11. A. N. Kolmogorov and S. V. Fomin,Elements of the Theory of Functions and Functional Analysis [in Russian] (3rd ed.), Nauka, Moscow (1972); English transl. prev. ed.: Vol. 1,Metric and Normal Spaces, Graylock, Rochester, NY (1957); Vol. 2,Measure. The Lebesque Integral, Hilbert Space Graylock, Albany, NY (1961).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Monocrystals, National Academy of Sciences of the Ukraine, Kharkov, The Ukraine

    Yu. P. Virchenko & O. L. Shpilinskaya

Authors
  1. Yu. P. Virchenko
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  2. O. L. Shpilinskaya
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.

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Virchenko, Y.P., Shpilinskaya, O.L. Random point fields with Markovian refinements and the geometry of fractally disordered media. Theor Math Phys 124, 1273–1285 (2000). https://doi.org/10.1007/BF02551004

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  • DOI: https://doi.org/10.1007/BF02551004

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