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April  2017, 37(4): 2161-2180. doi: 10.3934/dcds.2017093

## Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients

 1 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China 2 College of Mathematics and Physics, Chongqing, University of Posts and Telecommunications, Chongqing 400065, China

*Corresponding author: Daoyi Xu

Received  August 2015 Revised  November 2016 Published  December 2016

Fund Project: The first author is supported by National Natural Science Foundation of China grant No. 11271270 and the second author is supported by the Doctor Start-up Funding of Chongqing University of Posts and Telecommunications grant No. A2016-80

In this paper, we study the existence-uniqueness and exponential estimate of the pathwise mild solution of retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. Firstly, the existence-uniqueness of the maximal local pathwise mild solution are given by the generalized local Lipschitz conditions, which extend a classical Pazy theorem on PDEs. We assume neither that the noise is given in additive form or that it is a very simple multiplicative noise, nor that the drift coefficient is global Lipschitz continuous. Secondly, the existence-uniqueness of the global pathwise mild solution are given by establishing an integral comparison principle, which extends the classical Wintner theorem on ODEs. Thirdly, an exponential estimate for the pathwise mild solution is obtained by constructing a delay integral inequality. Finally, the results obtained are applied to a retarded stochastic infinite system and a stochastic partial functional differential equation. Combining some known results, we can obtain a random attractor, whose condition overcomes the disadvantage in existing results that the exponential converging rate is restricted by the maximal admissible value for the time delay.

Citation: Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093
##### References:
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Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, New York, 2007. Google Scholar [7] C. Cuevas, E. Hernándezb and M. Rabelo, The existence of solutions for impulsive neutral functional differential equations, Comp. Math. Appl., 58 (2009), 744-757. doi: 10.1016/j.camwa.2009.04.008. Google Scholar [8] J. Q. Duan, K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380. Google Scholar [9] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975. Google Scholar [10] C. Geiß and R. Manthey, Comparison theorems for stochastic differential equations in finite and infinite dimensions, Stochastic Process Appl., 53 (1994), 23-35. doi: 10.1016/0304-4149(94)90055-8. Google Scholar [11] J. K. Hale, Theorey of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar [12] X. Y. Han, W. X. Shen and S. F. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar [13] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc. , New York, 1964. Google Scholar [14] H. Kunita, Stochastic Flows and Stochastic Differential Equations, University Press, Cambridge, 1990. Google Scholar [15] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. Ⅱ, Academic Press, New York, 1969. Google Scholar [16] D. S. Li and D. Y. Xu, Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 1165-1181. doi: 10.4134/JKMS.2013.50.6.1165. Google Scholar [17] X. X. Liao and X. R. Mao, Exponential stability of stochastic delay interval systems, System Control Lett., 40 (2000), 171-181. doi: 10.1016/S0167-6911(00)00021-9. Google Scholar [18] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Pitman Monographs Ser. Pure Appl. Math. , 135 Chapman & Hall/CRC, 2006. Google Scholar [19] S. J. Long, L. Y. Teng and D. Y. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses, Statistics and Probability Lett., 82 (2012), 1699-1709. doi: 10.1016/j.spl.2012.05.018. Google Scholar [20] K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024. Google Scholar [21] X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar [22] V. M. Matrosov, The comparison principle with a Lyapunov vector-function Ⅰ-Ⅳ, Differential Equations, 4 (1968), 710-717; 893-900; 5 (1969), 853-864; 1596-1607.Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [24] G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [25] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992,185-192.Google Scholar [26] T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness, and asymptotic behavior of mild solution to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91. doi: 10.1006/jdeq.2001.4073. Google Scholar [27] L. S. Wang, Z. Zhang and Y. F. Wang, Stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters, Phys. Lett. A, 372 (2008), 3201-3209. doi: 10.1016/j.physleta.2007.07.090. Google Scholar [28] D. Y. Xu, Integro-differential equations and delay integral inequalities, Tohoku Math. J., 44 (1992), 365-378. doi: 10.2748/tmj/1178227303. Google Scholar [29] D. Y. Xu, Comparison theorems and vector Ⅴ-functions for stability of discrete systems, Proceedings of the Ninth Triennial World Congress of IFAC, 3 (1985), 1479-1482. Google Scholar [30] D. Y. Xu, Y. M. Huang and Z. G. Yang, Existence theory for periodic Markov process and stochastic functional differential equations, Disc. Contin. Dyn. Syst., 24 (2009), 1005-1023. doi: 10.3934/dcds.2009.24.1005. Google Scholar [31] D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Analysis, 108 (2014), 128-143. doi: 10.1016/j.na.2014.05.004. Google Scholar [32] D. Y. Xu, X. H. Wang and Z. G. Yang, Existence-uniqueness problems for infinte dimensional stochastic differential equations with delays, J. Appl. Anal. Comput., 2 (2012), 449-463. Google Scholar [33] D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equation, Sci. China Math., 56 (2013), 1169-1180. doi: 10.1007/s11425-012-4553-1. Google Scholar [34] D. Y. Xu and Z. C. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120. doi: 10.1016/j.jmaa.2004.10.040. Google Scholar [35] D. Y. Xu, Z. G. Yang and Y. M. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differential Equations, 245 (2008), 1681-1703. doi: 10.1016/j.jde.2008.03.029. Google Scholar [36] D. Y. Xu, H. Y. Zhao and H. Zhu, Global dynamics of Hopfield neural networks involving variable delays, Computer Math. Appl., 42 (2001), 39-45. doi: 10.1016/S0898-1221(01)00128-6. Google Scholar [37] X. H. Zhang, K. Wang and D. S. Li, Stochastic periodic solutions of stochastic differential equations driven by Lévy process, J. Math. Anal. Appl., 430 (2015), 231-242. doi: 10.1016/j.jmaa.2015.04.090. Google Scholar [38] H. Y. Zhao and N. Ding, Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays, Appl. Math. Comput., 183 (2006), 464-470. doi: 10.1016/j.amc.2006.05.087. Google Scholar

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar [2] P. W. Bates, K. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar [3] R. E. Bellman, Vector Lyapunov functions, J. Soc. Industr. Appl. Math. Ser. A Control., 1 (1962), 32-34. Google Scholar [4] H. Bessaih, M. J. Garrido-Atienza and B. Schmalfuss, Pathwise solutions and attractors for retarded SPDES with time smooth diffusion cofficients, Disc. Contin. Dyn. Syst., 34 (2014), 3945-3968. doi: 10.3934/dcds.2014.34.3945. Google Scholar [5] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property, Proc. R. Soc. Lond. A, 456 (2000), 1775-1802. doi: 10.1098/rspa.2000.0586. Google Scholar [6] P. L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, New York, 2007. Google Scholar [7] C. Cuevas, E. Hernándezb and M. Rabelo, The existence of solutions for impulsive neutral functional differential equations, Comp. Math. Appl., 58 (2009), 744-757. doi: 10.1016/j.camwa.2009.04.008. Google Scholar [8] J. Q. Duan, K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380. Google Scholar [9] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975. Google Scholar [10] C. Geiß and R. Manthey, Comparison theorems for stochastic differential equations in finite and infinite dimensions, Stochastic Process Appl., 53 (1994), 23-35. doi: 10.1016/0304-4149(94)90055-8. Google Scholar [11] J. K. Hale, Theorey of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar [12] X. Y. Han, W. X. Shen and S. F. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018. Google Scholar [13] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc. , New York, 1964. Google Scholar [14] H. Kunita, Stochastic Flows and Stochastic Differential Equations, University Press, Cambridge, 1990. Google Scholar [15] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. Ⅱ, Academic Press, New York, 1969. Google Scholar [16] D. S. Li and D. Y. Xu, Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 1165-1181. doi: 10.4134/JKMS.2013.50.6.1165. Google Scholar [17] X. X. Liao and X. R. Mao, Exponential stability of stochastic delay interval systems, System Control Lett., 40 (2000), 171-181. doi: 10.1016/S0167-6911(00)00021-9. Google Scholar [18] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Pitman Monographs Ser. Pure Appl. Math. , 135 Chapman & Hall/CRC, 2006. Google Scholar [19] S. J. Long, L. Y. Teng and D. Y. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses, Statistics and Probability Lett., 82 (2012), 1699-1709. doi: 10.1016/j.spl.2012.05.018. Google Scholar [20] K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024. Google Scholar [21] X. R. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar [22] V. M. Matrosov, The comparison principle with a Lyapunov vector-function Ⅰ-Ⅳ, Differential Equations, 4 (1968), 710-717; 893-900; 5 (1969), 853-864; 1596-1607.Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [24] G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [25] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992,185-192.Google Scholar [26] T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness, and asymptotic behavior of mild solution to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91. doi: 10.1006/jdeq.2001.4073. Google Scholar [27] L. S. Wang, Z. Zhang and Y. F. Wang, Stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters, Phys. Lett. A, 372 (2008), 3201-3209. doi: 10.1016/j.physleta.2007.07.090. Google Scholar [28] D. Y. Xu, Integro-differential equations and delay integral inequalities, Tohoku Math. J., 44 (1992), 365-378. doi: 10.2748/tmj/1178227303. Google Scholar [29] D. Y. Xu, Comparison theorems and vector Ⅴ-functions for stability of discrete systems, Proceedings of the Ninth Triennial World Congress of IFAC, 3 (1985), 1479-1482. Google Scholar [30] D. Y. Xu, Y. M. Huang and Z. G. Yang, Existence theory for periodic Markov process and stochastic functional differential equations, Disc. Contin. Dyn. Syst., 24 (2009), 1005-1023. doi: 10.3934/dcds.2009.24.1005. Google Scholar [31] D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Analysis, 108 (2014), 128-143. doi: 10.1016/j.na.2014.05.004. Google Scholar [32] D. Y. Xu, X. H. Wang and Z. G. Yang, Existence-uniqueness problems for infinte dimensional stochastic differential equations with delays, J. Appl. Anal. Comput., 2 (2012), 449-463. Google Scholar [33] D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equation, Sci. China Math., 56 (2013), 1169-1180. doi: 10.1007/s11425-012-4553-1. Google Scholar [34] D. Y. Xu and Z. C. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120. doi: 10.1016/j.jmaa.2004.10.040. Google Scholar [35] D. Y. Xu, Z. G. Yang and Y. M. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differential Equations, 245 (2008), 1681-1703. doi: 10.1016/j.jde.2008.03.029. Google Scholar [36] D. Y. Xu, H. Y. Zhao and H. Zhu, Global dynamics of Hopfield neural networks involving variable delays, Computer Math. Appl., 42 (2001), 39-45. doi: 10.1016/S0898-1221(01)00128-6. Google Scholar [37] X. H. Zhang, K. Wang and D. S. Li, Stochastic periodic solutions of stochastic differential equations driven by Lévy process, J. Math. Anal. Appl., 430 (2015), 231-242. doi: 10.1016/j.jmaa.2015.04.090. Google Scholar [38] H. Y. Zhao and N. Ding, Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays, Appl. Math. Comput., 183 (2006), 464-470. doi: 10.1016/j.amc.2006.05.087. Google Scholar
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