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Large time volume of the pinned Wiener sausage to second order
Author(s) -
McGillivray I.
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910460
Subject(s) - trace (psycholinguistics) , mathematics , dimension (graph theory) , order (exchange) , volume (thermodynamics) , function (biology) , energy (signal processing) , mathematical analysis , asymptotic expansion , mathematical physics , pure mathematics , quantum mechanics , physics , statistics , finance , philosophy , linguistics , evolutionary biology , economics , biology
We investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to a compact subset K in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\mathbb R}^d $\end{document} for d ⩾ 3. The structure of the asymptotic series is known: it depends strongly on whether the dimension d is odd or even; and the leading coefficient is given by the Newtonian capacity of K . In this article, we obtain detailed expressions for the second order coefficients. It is noteworthy that these coefficients feature novel potential‐theoretic terms that do not appear in the unpinned case. The proof exploits a trace formula of Krein. In the course of the argument, we derive a low‐energy expansion to second order for the Krein spectral shift function for an obstacle scattering system. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim