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Strong law of large numbers for the capacity of the Wiener sausage in dimension four

Abstract

We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This requires a delicate analysis of intersection probabilities between two independent Wiener sausages.

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Notes

  1. A mistake that Erdös and Taylor implicitly made in their pioneering work [7], and that Lawler corrected about twenty years later [11].

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Acknowledgements

We warmly thank the referee for his/her very careful reading and insightful comments, which greatly helped improve, clarify and correct the paper.

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Correspondence to Bruno Schapira.

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Asselah, A., Schapira, B. & Sousi, P. Strong law of large numbers for the capacity of the Wiener sausage in dimension four. Probab. Theory Relat. Fields 173, 813–858 (2019). https://doi.org/10.1007/s00440-018-0842-0

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  • DOI: https://doi.org/10.1007/s00440-018-0842-0

Keywords

  • Capacity
  • Wiener sausage
  • Law of large numbers

Mathematics Subject Classification

  • Primary 60F05
  • 60G50