Open Access
July 2011 Lack of strong completeness for stochastic flows
Xue-Mei Li, Michael Scheutzow
Ann. Probab. 39(4): 1407-1421 (July 2011). DOI: 10.1214/10-AOP585


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 two-dimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.


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Xue-Mei Li. Michael Scheutzow. "Lack of strong completeness for stochastic flows." Ann. Probab. 39 (4) 1407 - 1421, July 2011.


Published: July 2011
First available in Project Euclid: 5 August 2011

zbMATH: 1235.60066
MathSciNet: MR2857244
Digital Object Identifier: 10.1214/10-AOP585

Primary: 60H10
Secondary: 35B27 , 37C10

Keywords: Homogenization‎ , Stochastic differential equation , stochastic flow , strong completeness , weak completeness

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 4 • July 2011
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