The Annals of Probability

Lack of strong completeness for stochastic flows

Xue-Mei Li and Michael Scheutzow

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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.

Article information

Ann. Probab., Volume 39, Number 4 (2011), 1407-1421.

First available in Project Euclid: 5 August 2011

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 37C10: Vector fields, flows, ordinary differential equations 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]

Stochastic flow strong completeness weak completeness stochastic differential equation homogenization


Li, Xue-Mei; Scheutzow, Michael. Lack of strong completeness for stochastic flows. Ann. Probab. 39 (2011), no. 4, 1407--1421. doi:10.1214/10-AOP585.

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