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Forum Mathematicum

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Volume 29, Issue 2


A formula for the expected volume of the Wiener sausage with constant drift

Yuji Hamana / Hiroyuki Matsumoto
Published Online: 2016-06-14 | DOI: https://doi.org/10.1515/forum-2016-0039


We consider the Wiener sausage for a Brownian motion with a constant drift up to time t associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as κt(1+o[1]) for large t by a constant κ. The expression for κ is of a complicated form, but it converges to the known constant as the drift tends to 0.

Keywords: Wiener sausage; Brownian motion with drift; modified Bessel function

MSC 2010: 60J65; 44A10


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About the article

Received: 2016-02-15

Published Online: 2016-06-14

Published in Print: 2017-03-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 24540181

Award identifier / Grant number: 26400144

This work is partially supported by the Grant-in-Aid for Scientific Research (C) No. 24540181 and No. 26400144 of Japan Society for the Promotion of Science (JSPS).

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 369–381, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0039.

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