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Rubber bands, pursuit games and shy couplings
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Bramson, Maury, Burdzy, K. (Krzysztof) and Kendall, W. S. (2014) Rubber bands, pursuit games and shy couplings. Proceedings of the London Mathematical Society, 109 (1). pp. 121-160. doi:10.1112/plms/pdt067 ISSN 0024-6115.
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Official URL: http://dx.doi.org/10.1112/plms/pdt067
Abstract
In this paper, we consider pursuit–evasion and probabilistic consequences of some geometric notions for bounded and suitably regular domains in Euclidean space that are CAT(κ) for some κ>0 . These geometric notions are useful for analysing the related problems of (a) existence/ non-existence of successful evasion strategies for the Man in Lion and Man problems, and (b) existence/non-existence of shy couplings for reflected Brownian motions. They involve properties of rubber bands and the extent to which a loop in the domain in question can be deformed to a point without, in between, increasing its loop length. The existence of a stable rubber band will imply the existence of a successful evasion strategy but, if all loops in the domain are well-contractible, then no successful evasion strategy will exist and there can be no co-adapted shy coupling. For example, there can be no shy couplings in bounded and suitably regular star-shaped domains and so, in this setting, any two reflected Brownian motions must almost surely make arbitrarily close encounters as t→∞.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
Library of Congress Subject Headings (LCSH): | Brownian motion processes | ||||||||
Journal or Publication Title: | Proceedings of the London Mathematical Society | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0024-6115 | ||||||||
Official Date: | 10 July 2014 | ||||||||
Dates: |
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Volume: | 109 | ||||||||
Number: | 1 | ||||||||
Number of Pages: | 40 | ||||||||
Page Range: | pp. 121-160 | ||||||||
DOI: | 10.1112/plms/pdt067 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 28 December 2015 | ||||||||
Date of first compliant Open Access: | 28 December 2015 | ||||||||
Funder: | National Science Foundation (U.S.) (NSF), Poland. Ministerstwo Nauki i Szkolnictwa Wyższego [Ministry of Science and Higher Education] (MNiSW) | ||||||||
Grant number: | DMS-0906743 (NSF), DMS-1105668 (NSF), N N201 397137 (MNiSW) |
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