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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
##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). Google Scholar [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications,, Dynam. Stability Systems, 13 (1998), 265. Google Scholar [3] L. Arnold and I. Chueshov, Cooperative random and stochastic differential equations,, Discrete Continuous Dynam. Systems - A, 7 (2001), 1. Google Scholar [4] B. Bergé, I. Chueshov and P. Vuillermot, On the behavior of solutions to certain parabolic SPDE's driven by Wiener processes,, Stochastic Process. Appl., 92 (2001), 237. doi: 10.1016/S0304-4149(00)00082-X. Google Scholar [5] I. Chueshov, Order-preserving random dynamical systems generated by a class of coupled stochastic semilinear parabolic equations,, in, (2000), 711. Google Scholar [6] I. Chueshov, "Monotone Random Systems: Theory and Applications,", Lecture Notes in Mathematics, 1779 (2002). doi: 10.1007/b83277. Google Scholar [7] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems: An International Journal, 19 (2004), 127. doi: 10.1080/1468936042000207792. Google Scholar [8] I. Chueshov and B. Schmalfuss, Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions,, Discrete Continuous Dynam. Systems - A, 18 (2007), 315. doi: 10.3934/dcds.2007.18.315. Google Scholar [9] I. Chueshov and P. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case,, Probab. Theory Rel. Fields, 112 (1998), 149. doi: 10.1007/s004400050186. Google Scholar [10] I. Chueshov and P. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case,, Stoch. Anal. Appl., 18 (2000), 581. doi: 10.1080/07362990008809687. Google Scholar [11] I. Chueshov and P. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations,, Stoch. Anal. Appl., 22 (2004), 1421. doi: 10.1081/SAP-200029487. Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Relat. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [13] A. Es-Sarhir, M. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term,, Diff. Integral Equations, 23 (2010), 189. Google Scholar [14] M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations,, Probab. Theory Relat. Fields, 149 (2011), 223. doi: 10.1007/s00440-009-0250-6. Google Scholar [15] P. Imkeller and M. Scheutzow, On the spatial asymptotic behaviour of stochastic flows in Euclidean space,, Ann. Probab., 27 (1999), 109. doi: 10.1214/aop/1022677255. Google Scholar [16] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation,, J. Math. Kyoto Univ., 4 (1964), 1. Google Scholar [17] M. Krasnosel'skii, "Positive Solutions of Operator Equations,", Noordhoff Ltd. Groningen, (1964). Google Scholar [18] M. Krasnosel'skii, Je. A. Lifshits and A. Sobolev, "Positive Linear Systems. The Method of Positive Operators,", Sigma Series in Applied Mathematics, 5 (1989). Google Scholar [19] H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (1990). Google Scholar [20] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. AMS, 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [21] R. Martin, Jr. and H. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. Google Scholar [22] S. Mohammed, Nonlinear flows of stochastic linear delay equations,, Stochastics, 17 (1986), 207. doi: 10.1080/17442508608833390. Google Scholar [23] S. Mohammed, The Lyapunov spectrum and stable manifolds for stochastic linear delay equations,, Stochastics and Stochastic Reports, 29 (1990), 89. Google Scholar [24] S. Mohammed and M. Scheutzow, Spatial estimates for stochastic flows in Euclidean space,, Ann. Probab., 26 (1998), 56. doi: 10.1214/aop/1022855411. Google Scholar [25] S. Mohammed and M. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow,, J. Functional Anal., 205 (2003), 271. doi: 10.1016/j.jfa.2002.04.001. Google Scholar [26] G. Ochs, "Weak Random Attractors,", Institut für Dynamische Systeme, (1999). Google Scholar [27] M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory,, Stochastics, 12 (1984), 41. doi: 10.1080/17442508408833294. Google Scholar [28] M. Scheutzow, Comparison of various concepts of a random attractor: A case study,, Arch. Math., 78 (2002), 233. doi: 10.1007/s00013-002-8241-1. Google Scholar [29] B. Schmalfuss, Backward cocycles and attractors for stochastic differential equations,, in, (1992), 185. Google Scholar [30] G. Seifert, Positively invariant closed sets for systems of delay differential equations,, J. Diff. Eqs., 22 (1976), 292. Google Scholar [31] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995). Google Scholar [32] T. Tibor, H.-O. Walther and J. Wu, The structure of an attracting set defined by delayed and monotone positive feedback,, CWI Quarterly, 12 (1999), 315. Google Scholar

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##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). Google Scholar [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications,, Dynam. Stability Systems, 13 (1998), 265. Google Scholar [3] L. Arnold and I. Chueshov, Cooperative random and stochastic differential equations,, Discrete Continuous Dynam. Systems - A, 7 (2001), 1. Google Scholar [4] B. Bergé, I. Chueshov and P. Vuillermot, On the behavior of solutions to certain parabolic SPDE's driven by Wiener processes,, Stochastic Process. Appl., 92 (2001), 237. doi: 10.1016/S0304-4149(00)00082-X. Google Scholar [5] I. Chueshov, Order-preserving random dynamical systems generated by a class of coupled stochastic semilinear parabolic equations,, in, (2000), 711. Google Scholar [6] I. Chueshov, "Monotone Random Systems: Theory and Applications,", Lecture Notes in Mathematics, 1779 (2002). doi: 10.1007/b83277. Google Scholar [7] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems: An International Journal, 19 (2004), 127. doi: 10.1080/1468936042000207792. Google Scholar [8] I. Chueshov and B. Schmalfuss, Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions,, Discrete Continuous Dynam. Systems - A, 18 (2007), 315. doi: 10.3934/dcds.2007.18.315. Google Scholar [9] I. Chueshov and P. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case,, Probab. Theory Rel. Fields, 112 (1998), 149. doi: 10.1007/s004400050186. Google Scholar [10] I. Chueshov and P. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case,, Stoch. Anal. Appl., 18 (2000), 581. doi: 10.1080/07362990008809687. Google Scholar [11] I. Chueshov and P. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations,, Stoch. Anal. Appl., 22 (2004), 1421. doi: 10.1081/SAP-200029487. Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Relat. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [13] A. Es-Sarhir, M. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term,, Diff. Integral Equations, 23 (2010), 189. Google Scholar [14] M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations,, Probab. Theory Relat. Fields, 149 (2011), 223. doi: 10.1007/s00440-009-0250-6. Google Scholar [15] P. Imkeller and M. Scheutzow, On the spatial asymptotic behaviour of stochastic flows in Euclidean space,, Ann. Probab., 27 (1999), 109. doi: 10.1214/aop/1022677255. Google Scholar [16] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation,, J. Math. Kyoto Univ., 4 (1964), 1. Google Scholar [17] M. Krasnosel'skii, "Positive Solutions of Operator Equations,", Noordhoff Ltd. Groningen, (1964). Google Scholar [18] M. Krasnosel'skii, Je. A. Lifshits and A. Sobolev, "Positive Linear Systems. The Method of Positive Operators,", Sigma Series in Applied Mathematics, 5 (1989). Google Scholar [19] H. Kunita, "Stochastic Flows and Stochastic Differential Equations,", Cambridge Studies in Advanced Mathematics, 24 (1990). Google Scholar [20] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. AMS, 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [21] R. Martin, Jr. and H. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. Google Scholar [22] S. Mohammed, Nonlinear flows of stochastic linear delay equations,, Stochastics, 17 (1986), 207. doi: 10.1080/17442508608833390. Google Scholar [23] S. Mohammed, The Lyapunov spectrum and stable manifolds for stochastic linear delay equations,, Stochastics and Stochastic Reports, 29 (1990), 89. Google Scholar [24] S. Mohammed and M. Scheutzow, Spatial estimates for stochastic flows in Euclidean space,, Ann. Probab., 26 (1998), 56. doi: 10.1214/aop/1022855411. Google Scholar [25] S. Mohammed and M. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow,, J. Functional Anal., 205 (2003), 271. doi: 10.1016/j.jfa.2002.04.001. Google Scholar [26] G. Ochs, "Weak Random Attractors,", Institut für Dynamische Systeme, (1999). Google Scholar [27] M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory,, Stochastics, 12 (1984), 41. doi: 10.1080/17442508408833294. Google Scholar [28] M. Scheutzow, Comparison of various concepts of a random attractor: A case study,, Arch. Math., 78 (2002), 233. doi: 10.1007/s00013-002-8241-1. Google Scholar [29] B. Schmalfuss, Backward cocycles and attractors for stochastic differential equations,, in, (1992), 185. Google Scholar [30] G. Seifert, Positively invariant closed sets for systems of delay differential equations,, J. Diff. Eqs., 22 (1976), 292. Google Scholar [31] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Mathematical Surveys and Monographs, 41 (1995). Google Scholar [32] T. Tibor, H.-O. Walther and J. Wu, The structure of an attracting set defined by delayed and monotone positive feedback,, CWI Quarterly, 12 (1999), 315. Google Scholar
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