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A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations
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 and 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, Springer-Verlag, Berlin, 1998. [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dynam. Stability Systems, 13 (1998), 265-280. [3] L. Arnold and I. Chueshov, Cooperative random and stochastic differential equations, Discrete Continuous Dynam. Systems - A, 7 (2001), 1-33. [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), 237-263. doi: 10.1016/S0304-4149(00)00082-X. [5] I. Chueshov, Order-preserving random dynamical systems generated by a class of coupled stochastic semilinear parabolic equations, 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. [6] I. Chueshov, "Monotone Random Systems: Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [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), 127-144. doi: 10.1080/1468936042000207792. [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), 315-338. doi: 10.3934/dcds.2007.18.315. [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), 149-202. doi: 10.1007/s004400050186. [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), 581-615. doi: 10.1080/07362990008809687. [11] I. Chueshov and P. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations, Stoch. Anal. Appl., 22 (2004), 1421-1486. doi: 10.1081/SAP-200029487. [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [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), 189-200. [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), 223-259. doi: 10.1007/s00440-009-0250-6. [15] P. Imkeller and M. Scheutzow, On the spatial asymptotic behaviour of stochastic flows in Euclidean space, Ann. Probab., 27 (1999), 109-129. doi: 10.1214/aop/1022677255. [16] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75. [17] M. Krasnosel'skii, "Positive Solutions of Operator Equations," Noordhoff Ltd. Groningen, 1964. [18] M. Krasnosel'skii, Je. A. Lifshits and A. Sobolev, "Positive Linear Systems. The Method of Positive Operators," Sigma Series in Applied Mathematics, 5, Heldermann-Verlag, Berlin, 1989. [19] H. Kunita, "Stochastic Flows and Stochastic Differential Equations," Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [20] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. AMS, 321 (1990), 1-44. doi: 10.2307/2001590. [21] R. Martin, Jr. and H. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35. [22] S. Mohammed, Nonlinear flows of stochastic linear delay equations, Stochastics, 17 (1986), 207-213. doi: 10.1080/17442508608833390. [23] S. Mohammed, The Lyapunov spectrum and stable manifolds for stochastic linear delay equations, Stochastics and Stochastic Reports, 29 (1990), 89-131. [24] S. Mohammed and M. Scheutzow, Spatial estimates for stochastic flows in Euclidean space, Ann. Probab., 26 (1998), 56-77. doi: 10.1214/aop/1022855411. [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), 271-305. doi: 10.1016/j.jfa.2002.04.001. [26] G. Ochs, "Weak Random Attractors," Institut für Dynamische Systeme, Universität Bremen, Report 449, 1999. [27] M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory, Stochastics, 12 (1984), 41-80. doi: 10.1080/17442508408833294. [28] M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Arch. Math., 78 (2002), 233-240. doi: 10.1007/s00013-002-8241-1. [29] B. Schmalfuss, Backward cocycles and attractors for stochastic differential equations, 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. [30] G. Seifert, Positively invariant closed sets for systems of delay differential equations, J. Diff. Eqs., 22 (1976), 292-304. [31] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island, 1995. [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), 315-327.

show all references

##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dynam. Stability Systems, 13 (1998), 265-280. [3] L. Arnold and I. Chueshov, Cooperative random and stochastic differential equations, Discrete Continuous Dynam. Systems - A, 7 (2001), 1-33. [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), 237-263. doi: 10.1016/S0304-4149(00)00082-X. [5] I. Chueshov, Order-preserving random dynamical systems generated by a class of coupled stochastic semilinear parabolic equations, 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. [6] I. Chueshov, "Monotone Random Systems: Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [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), 127-144. doi: 10.1080/1468936042000207792. [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), 315-338. doi: 10.3934/dcds.2007.18.315. [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), 149-202. doi: 10.1007/s004400050186. [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), 581-615. doi: 10.1080/07362990008809687. [11] I. Chueshov and P. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations, Stoch. Anal. Appl., 22 (2004), 1421-1486. doi: 10.1081/SAP-200029487. [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [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), 189-200. [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), 223-259. doi: 10.1007/s00440-009-0250-6. [15] P. Imkeller and M. Scheutzow, On the spatial asymptotic behaviour of stochastic flows in Euclidean space, Ann. Probab., 27 (1999), 109-129. doi: 10.1214/aop/1022677255. [16] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75. [17] M. Krasnosel'skii, "Positive Solutions of Operator Equations," Noordhoff Ltd. Groningen, 1964. [18] M. Krasnosel'skii, Je. A. Lifshits and A. Sobolev, "Positive Linear Systems. The Method of Positive Operators," Sigma Series in Applied Mathematics, 5, Heldermann-Verlag, Berlin, 1989. [19] H. Kunita, "Stochastic Flows and Stochastic Differential Equations," Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. [20] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. AMS, 321 (1990), 1-44. doi: 10.2307/2001590. [21] R. Martin, Jr. and H. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35. [22] S. Mohammed, Nonlinear flows of stochastic linear delay equations, Stochastics, 17 (1986), 207-213. doi: 10.1080/17442508608833390. [23] S. Mohammed, The Lyapunov spectrum and stable manifolds for stochastic linear delay equations, Stochastics and Stochastic Reports, 29 (1990), 89-131. [24] S. Mohammed and M. Scheutzow, Spatial estimates for stochastic flows in Euclidean space, Ann. Probab., 26 (1998), 56-77. doi: 10.1214/aop/1022855411. [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), 271-305. doi: 10.1016/j.jfa.2002.04.001. [26] G. Ochs, "Weak Random Attractors," Institut für Dynamische Systeme, Universität Bremen, Report 449, 1999. [27] M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory, Stochastics, 12 (1984), 41-80. doi: 10.1080/17442508408833294. [28] M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Arch. Math., 78 (2002), 233-240. doi: 10.1007/s00013-002-8241-1. [29] B. Schmalfuss, Backward cocycles and attractors for stochastic differential equations, 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. [30] G. Seifert, Positively invariant closed sets for systems of delay differential equations, J. Diff. Eqs., 22 (1976), 292-304. [31] H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island, 1995. [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), 315-327.
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