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August  2013, 18(6): 1533-1554. doi: 10.3934/dcdsb.2013.18.1533

Invariance and monotonicity for stochastic delay differential equations

1. 

Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61022, Ukraine

2. 

Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

Received  January 2012 Revised  March 2012 Published  March 2013

We study invariance and monotonicity properties of Kunita-type sto-chastic differential equations in $\mathbb{R}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\mathbb{R}^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.
Citation: Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
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show all references

References:
[1]

Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar

[2]

Dynam. Stability Systems, 13 (1998), 265-280.  Google Scholar

[3]

Discrete Continuous Dynam. Systems - A, 7 (2001), 1-33.  Google Scholar

[4]

Stochastic Process. Appl., 92 (2001), 237-263. doi: 10.1016/S0304-4149(00)00082-X.  Google Scholar

[5]

in "International Conference on Differential Equations, Vol. 1, 2" (eds. B. Fiedler, K. Gröger and J. Sprekels) (Berlin, 1999), World Scientific Publ., River Edge, NJ, (2000), 711-716.  Google Scholar

[6]

Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[7]

Dynamical Systems: An International Journal, 19 (2004), 127-144. doi: 10.1080/1468936042000207792.  Google Scholar

[8]

Discrete Continuous Dynam. Systems - A, 18 (2007), 315-338. doi: 10.3934/dcds.2007.18.315.  Google Scholar

[9]

Probab. Theory Rel. Fields, 112 (1998), 149-202. doi: 10.1007/s004400050186.  Google Scholar

[10]

Stoch. Anal. Appl., 18 (2000), 581-615. doi: 10.1080/07362990008809687.  Google Scholar

[11]

Stoch. Anal. Appl., 22 (2004), 1421-1486. doi: 10.1081/SAP-200029487.  Google Scholar

[12]

Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar

[13]

Diff. Integral Equations, 23 (2010), 189-200.  Google Scholar

[14]

Probab. Theory Relat. Fields, 149 (2011), 223-259. doi: 10.1007/s00440-009-0250-6.  Google Scholar

[15]

Ann. Probab., 27 (1999), 109-129. doi: 10.1214/aop/1022677255.  Google Scholar

[16]

J. Math. Kyoto Univ., 4 (1964), 1-75.  Google Scholar

[17]

Noordhoff Ltd. Groningen, 1964.  Google Scholar

[18]

Sigma Series in Applied Mathematics, 5, Heldermann-Verlag, Berlin, 1989.  Google Scholar

[19]

Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990.  Google Scholar

[20]

Trans. AMS, 321 (1990), 1-44. doi: 10.2307/2001590.  Google Scholar

[21]

J. Reine Angew. Math., 413 (1991), 1-35.  Google Scholar

[22]

Stochastics, 17 (1986), 207-213. doi: 10.1080/17442508608833390.  Google Scholar

[23]

Stochastics and Stochastic Reports, 29 (1990), 89-131. Google Scholar

[24]

Ann. Probab., 26 (1998), 56-77. doi: 10.1214/aop/1022855411.  Google Scholar

[25]

J. Functional Anal., 205 (2003), 271-305. doi: 10.1016/j.jfa.2002.04.001.  Google Scholar

[26]

Institut für Dynamische Systeme, Universität Bremen, Report 449, 1999. Google Scholar

[27]

Stochastics, 12 (1984), 41-80. doi: 10.1080/17442508408833294.  Google Scholar

[28]

Arch. Math., 78 (2002), 233-240. doi: 10.1007/s00013-002-8241-1.  Google Scholar

[29]

in "International Seminar on Applied Mathematics - Nonlinear Dynamics: Attractor Approximation and Global Behaviour" (eds. V. Reitmann, T. Riedrich and N. Koksch), Teubner, (1992), 185-192. Google Scholar

[30]

J. Diff. Eqs., 22 (1976), 292-304.  Google Scholar

[31]

Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island, 1995.  Google Scholar

[32]

CWI Quarterly, 12 (1999), 315-327. Google Scholar

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