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Lack of strong completeness for stochastic flows
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Li, XM. (XueMei), 1964 and Scheutzow, Michael. (2011) Lack of strong completeness for stochastic flows. The Annals of Probability, Vol.39 (No.4). pp. 14071421. ISSN 00911798

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Official URL: http://dx.doi.org/10.1214/10AOP585
Abstract
It is well known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If, in addition, the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition x, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently of x, then the maximal flow is called strongly complete. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a twodimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Stochastic processes, Stochastic differential equations 
Journal or Publication Title:  The Annals of Probability 
Publisher:  Institute of Mathematical Statistics 
ISSN:  00911798 
Official Date:  July 2011 
Volume:  Vol.39 
Number:  No.4 
Page Range:  pp. 14071421 
Identification Number:  10.1214/10AOP585 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  Engineering and Physical Sciences Research Council (EPSRC) 
Grant number:  EP/E058124/1 (EPSRC) 
References:  [1] BAXENDALE, P. (1980/81).Wiener processes on manifolds of maps. Proc. Roy. Soc. Edinburgh 
URI:  http://wrap.warwick.ac.uk/id/eprint/38510 
Data sourced from Thomson Reuters' Web of Knowledge
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