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On the Volume of the Intersection of Two Independent Wiener Sausages

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Abstract

The expected volume of intersection of two independent Wiener sausages in ℝm, m ≥ 3, up to time t, and associated to non-polar, compact sets K 1 and K 2 respectively, is obtained in the limit of large t.

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References

  1. Albeverio, S., Zhou, X.Y.: Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45, 195–237 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. van den Berg, M.: On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals. J. Funct. Anal. 222, 114–128 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. van den Berg, M.: On the volume of intersection of three independent Wiener sausages. Annales de L’Institut Henri Poincaré, B. Probabilités et Statistiques 46, 313–337 (2010)

    Article  MATH  Google Scholar 

  4. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Oxford University Press, Oxford (1992)

    Google Scholar 

  5. Fernández, R., Fröhlich, J., Sokal, A.D.: Random walks, critical phenomena and triviality in quantum field theory. In: Texts and Monographs in Physics. Springer, New York (1992)

    Google Scholar 

  6. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, San Diego (1994)

    MATH  Google Scholar 

  7. Lawler, G.: Intersections of random walks. In: Probability and its Applications. Birkhäuser, Boston (1991)

    Google Scholar 

  8. Le Gall, J.-F.: Sur une conjecture de M. Kac. Probab. Theory Relat. Fields 78, 389–402 (1988)

    Article  MATH  Google Scholar 

  9. Le Gall, J.-F.: Wiener sausage and self-intersection local time. J. Funct. Anal. 88, 299–341 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser, Boston, (1993)

    MATH  Google Scholar 

  11. Port, S.C.: Asymptotic expansions for the expected volume of a stable sausage. Ann. Probab. 18, 492–523 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Port, S.C.: Spitzer’s formula involving capacity. Prog. Probab. 28, 373–388 (1991)

    MathSciNet  Google Scholar 

  13. Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic, New York (1967)

    Google Scholar 

  14. Spitzer, F.: Electrostatic capacity, heat flow and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb. 3, 110–121 (1964)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Michiel van den Berg

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  1. Michiel van den Berg
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Correspondence to Michiel van den Berg.

Additional information

Research supported by The Leverhulme Trust, Research Fellowship 2008/0368.

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van den Berg, M. On the Volume of the Intersection of Two Independent Wiener Sausages. Potential Anal 34, 57–79 (2011). https://doi.org/10.1007/s11118-010-9181-1

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  • DOI: https://doi.org/10.1007/s11118-010-9181-1

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