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Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-21826

van den Berg, M; Bolthausen, E; den Hollander, F (2004). On the volume of the intersection of two Wiener sausages. Annals of Mathematics. Second Series, 159(2):741-782.

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## Abstract

For , let be the -neighbourhoods of the th copy of a standard Brownian motion in starting at 0, until time . The authors prove large deviations results about , for , and suggest extensions applicable to , the volume of the intersection of sausages.

In particular, for , -I_d^(c)t (here is the Newtonian capacity of the ball of radius ). A similar result holds for with replaced by and replaced by . The sizes of the large deviations come from the asymptotic value of the expected volume of a single Wiener sausage. A variational representation is derived for , and the authors also investigate the dependence of on for different values of .

The work is motivated by the desire to address a number of open problems arising in the discrete setting from the study of the tail of the distribution of the intersection of the ranges of two independent random walks in (in such cases no exact rate constant is known).

The results in the paper draw on ideas and techniques developed by the authors to handle large deviations for the volume of a single Wiener sausage.

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Wiener sausages, intersection volume, large deviations, variational problems, Sobolev inequalities. English 2004 21 Apr 2010 15:15 28 Nov 2013 01:34 Mathematical Sciences Publishers 0003-486X http://annals.math.princeton.edu/annals/2004/159-2/p06.xhtml Web of Science®. Times Cited: 5Google Scholar™