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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 
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 Sect. A 87 127–152. MR600452 [2] BLAGOVEŠ ˇ CENSKI˘I, J. N. and FRE˘IDLIN, M. I. (1961). Some properties of diffusion processes depending on a parameter. Dokl. Akad. Nauk SSSR 138 508–511. MR0139196 [3] CARVERHILL, A. P. (1981). A pair of stochastic dynamical systems which have the same infinitesimal generator, but of which one is strongly complete and the other is not. Univ. Warwick. Preprint. [4] CARVERHILL, A. P. and ELWORTHY, K. D. (1983). Flows of stochastic dynamical systems: The functional analytic approach. Z. Wahrsch. Verw. Gebiete 65 245–267. MR722131 [5] CLARK, J. M. C. (1973). An introduction to stochastic differential equations on manifolds. In Geometric Methods in Systems Theory (D. Q. Mayne and R. W. Brockett, eds.) 131–149. Reidel, Dordrecht. [6] ELWORTHY, K. D. (1982). Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Note Series 70. Cambridge Univ. Press, Cambridge. MR675100 [7] ELWORTHY, K. D. (1978). Stochastic dynamical systems and their flows. In Stochastic Analysis (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978) 79–95. Academic Press, New York. MR517235 [8] FANG, S., IMKELLER, P. and ZHANG, T. (2007). Global flows for stochastic differential equations without global Lipschitz conditions. Ann. Probab. 35 180–205. MR2303947 [9] JACOD, J. and SHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. MR1943877 [10] KUNITA, H. (1981). On the decomposition of solutions of stochastic differential equations. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 213–255. Springer, Berlin. MR620992 [11] KUNITA, H. (1984). Stochastic differential equations and stochastic flows of homeomorphisms. In Stochastic Analysis and Applications. Adv. Probab. Related Topics 7 269–291. Dekker, New York. MR776984 [12] KUNITA, H. (1986). Lectures on Stochastic Flows and Applications. Tata Institute of Fundamental Research Lectures on Mathematics and Physics 78. Springer, Berlin. MR867686 [13] KUNITA, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge. MR1070361 [14] LI, X.M. (1994). Strong pcompleteness of stochastic differential equations and the existence of smooth flows on noncompact manifolds. Probab. Theory Related Fields 100 485–511. MR1305784 [15] MOHAMMED, S. and SCHEUTZOW, M. (2003). The stable manifold theorem for nonlinear stochastic systems with memory I: Existence of the semiflow. J. Funct. Anal. 205 271– 305. MR2017689 
URI:  http://wrap.warwick.ac.uk/id/eprint/38510 
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