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Hausdorff dimension of the sample path of the Westwater process

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Abstract

In this paper, we prove that the Hausdorff dimension of the sample path of the three dimensional coordinate process is exactly two a.e. −ν(g), where ν(g) is the polymer measure constructed by Westwater. Furthermore, we prove that the sample path of the Westwater process has also zero Hausdorff 2-measure. This answers a problem suggested by E. Nelson.

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References

  1. Adler, R. J.,The Geometry of Random Fields, Wiley Series in Prob. Math. Stat., Chichester, 1981.

  2. Adler, R. J.,The uniform dimension of the level sets of a Brownian sheet, Ann. Prob.,6 (1978), 509–515.

    MATH  Google Scholar 

  3. Ciesielski, Z., Taylor, S. J.,First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Am. Math. Soc.,103 (1962), 434–450.

    Article  MATH  MathSciNet  Google Scholar 

  4. Ehm, W.,Sample function properties of multi-parameter stable processes, Z.W. Verw. Gebiete,56 (1981), 195–228.

    Article  MATH  MathSciNet  Google Scholar 

  5. Kusuoka, S.,The path property of Edward's model for long polymer chains in three dimensions, Res. Notes in Math. Pitman, London, 1985, 48–65.

    Google Scholar 

  6. Lévy, P.,La measure de Hausdorff de la courbe du mouvement brownien, Giorn, Ist, Ital. Attuari,16 (1953), 1–37.

    MathSciNet  Google Scholar 

  7. Nelson, E.,A remark on the polymer problem in four dimensions, Studies in Appl. Math. Adv. in Math. Supp. Stud.,8 (1983), Ed. V.V. Guillemin, 1–5.

    Google Scholar 

  8. Stroock, D.W., Varadhan, S. R.S.,Multidimensional Diffusion Processes, Springer-Verlag, 1979.

  9. Symanzik, K.,Euclidean quantum field theory, In “Local Quantum Theory” (R. Jost (ed.) Academic Press, New York, 1969.

    Google Scholar 

  10. Westwater, M. J.,On Edward's model for long polymer chains, Comm. Math. Phys.,72 (1980), 131–174.

    Article  MATH  MathSciNet  Google Scholar 

  11. Westwater, M. J.,On Edward's model for polymer chains III, Borel summability, Comm. Math. Phys.,84 (1982), 459–470.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Zhou Xian-Yin

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  1. Zhou Xian-Yin
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Research supported in part by the National Natural Science Foundation of China

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Xian-Yin, Z. Hausdorff dimension of the sample path of the Westwater process. Acta Mathematica Sinica 8, 26–45 (1992). https://doi.org/10.1007/BF02595017

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  • DOI: https://doi.org/10.1007/BF02595017

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