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Random-Path Intersections in Geometry, Probability and Physics

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Stochastic Processes and their Applications

Part of the book series: Mathematics and Its Applications (Soviet Series) ((MAIA,volume 61))

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Abstract

Recent progress in mathematical physics has generated an interest in certain random intersections.

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References

  1. Aizenman, M. Geometric Analysis of φ4 Fields and Ising Models. Parts I and II. Comm. Math. Phys. 86 (1982) 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  2. Berbee, H. Personal communication, 1982.

    Google Scholar 

  3. Daley, D.J. On a class of renewal functions. Math. Proc. Camb. Phil. Soc. 61 (1965) 519–526.

    Article  MathSciNet  MATH  Google Scholar 

  4. Dynkin, E.B. Local times and quantum fields. In Seminar on Stochastic Processes eds. E. Cinlar, K.L. Chung and R.K. Getoor, Birkhäuser. D

    Google Scholar 

  5. Edwards, S.F. The statistical mechanics of polymers with excluded volume. Proc. Phys. Sci. 85(1965) 613–624.

    Article  MATH  Google Scholar 

  6. Feller, W. An Introduction to Probability Theory and its Applications. Vol. 1, 3rd Edition. Wiley, New York, 1966.

    MATH  Google Scholar 

  7. Frostman, O. Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Thesis, Lund University, 1935.

    Google Scholar 

  8. Geman, D., J. Horowitz and J. Rosen. A local time analysis of intersections of brownian paths in the plane.Ann. Probab.12 (1984) 86–107.

    Article  MathSciNet  MATH  Google Scholar 

  9. Hawkes, J. Hausdorff measure, entropy and the independence of small sets. Proc. London Math. Soc. 28 (1974) 700–724.

    Article  MathSciNet  MATH  Google Scholar 

  10. Hawkes, J. Intersections of Markov random sets. Z. Wahrscheinlichkeitstheorie 37 (1977) 243–251.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hawkes, J. Multiple points for symmetric Lévy processes. Math. Proc. Camb. Phil. Soc. 83(1978) 83–90.

    MathSciNet  MATH  Google Scholar 

  12. Hawkes, J. Some geometric aspects of potential theory. In Proc. Swansea Conf. Stochastic Analysis and Applications eds. A. Truman and D. Williams, Lect. Notes Math. 1095 130–154, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  13. Hawkes, J. Fourier methods in the geometry of small sets. In preparation.

    Google Scholar 

  14. Hoffman-Jorgensen, J. Markov sets. Math. Scand. 24(1969) 145–166.

    MathSciNet  Google Scholar 

  15. Horowitz, J. Semilinear Markov processes, subordinators and renewal theory. Z. Wahrscheinlichkeitstheorie 24 (1972)167–193.

    Google Scholar 

  16. Kesten, H. Hitting probabilities of single points for processes with stationary independent increments. Mem. Amer. Math. Soc. 93 (1969).

    Google Scholar 

  17. Kingman, J.F.C. Regenerative phenomena. Wiley, New York, 1972.

    Google Scholar 

  18. Kusuoka, S. On the path property of Edwards’ model for long polymer chains in three dimensions. Preprint, 1984.

    Google Scholar 

  19. Lamperti, J. On the coefficients of reciprocal power series. Amer. Math. Monthly 65(1958) 90–94.

    Article  MathSciNet  MATH  Google Scholar 

  20. le jan, Y. Stochastic flows of diffeomorphisms. Talk, 1st BiBoS Symposium, Bielefeld, 1984.

    Google Scholar 

  21. Lévy, P. Une hierarchie des probabilités plus ou moins nulles, application â certains nuages de points. Enseignement Math. (2) 15 (1969) 217–225.

    Google Scholar 

  22. Lévy, P. Esquisse d’un calcul des probabilités plus ou moins nulles. Rev. Roumaine Math. Pures App l14. (1969) 813–818.

    MATH  Google Scholar 

  23. Marstrand, J.M. Some fundamental geometric properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3) 4 (1954) 257–302.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rosen, J. Self-intersection of random fields. Ann. Probab.12 (1984) 108–119.

    Article  MathSciNet  MATH  Google Scholar 

  25. Rosen, J. A local time approach to the self-intersections of brownian paths in space. Comm. Math. Phys. 88(1983) 327–338.

    Article  MATH  Google Scholar 

  26. Smith, W.L. Infinitesimal renewal processes. In Contributions to Probability and Statistics ed. I. Olkin, pp. 396–413. Stanford University Press, 1960.

    Google Scholar 

  27. Symanzik, K. Euclidean quantum field theory. In Local Quantum Theory ed. R. Jost, pp. 152–219. Academic Press, New York, 1969.

    Google Scholar 

  28. Szasz, D. A problem of two lifts. Ann. Probab. 5 (1977) 550–559.

    Article  MathSciNet  MATH  Google Scholar 

  29. Westwater, J. On Edwards’ model for long polymer chains. Comm. Math. Phys. 72(1980) 131–174.

    Article  MathSciNet  MATH  Google Scholar 

  30. Westwater, J. On Edwards’ model for polymer chains: II. The self-consistent potential. Comm. Math. Phys. 79(1981) 53–73.

    Article  MathSciNet  MATH  Google Scholar 

  31. Westwater, J. On Edwards’ model for polymer chains: III. Borel summability. Comm. Math. Phys. 84(1982) 459–470.

    Article  MathSciNet  MATH  Google Scholar 

  32. WolperT, R.L. Wiener path intersections and local time. J. Funct. Anal. 30(1978) 329–340.

    Article  MathSciNet  MATH  Google Scholar 

  33. Wolpert, R.L. Local time and a particle picture for Euclidean field theory. J. Funct. Anal. 30(1978) 341–357.

    Article  MathSciNet  MATH  Google Scholar 

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© 1990 Kluwer Academic Publishers

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Hawkes, J. (1990). Random-Path Intersections in Geometry, Probability and Physics. In: Albeverio, S., Streit, L., Blanchard, P. (eds) Stochastic Processes and their Applications. Mathematics and Its Applications (Soviet Series), vol 61. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2117-7_11

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  • DOI: https://doi.org/10.1007/978-94-009-2117-7_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7452-0

  • Online ISBN: 978-94-009-2117-7

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