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On nonuniform dichotomy for stochastic skew-evolution semiflows in Hilbert spaces
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 879-887
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In this paper we study a general concept of nonuniform exponential dichotomy in mean square for stochastic skew-evolution semiflows in Hilbert spaces. We obtain a variant for the stochastic case of some well-known results, of the deterministic case, due to R. Datko: Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3(1972), 428–445. Our approach is based on the extension of some techniques used in the deterministic case for the study of asymptotic behavior of skew-evolution semiflows in Banach spaces.
In this paper we study a general concept of nonuniform exponential dichotomy in mean square for stochastic skew-evolution semiflows in Hilbert spaces. We obtain a variant for the stochastic case of some well-known results, of the deterministic case, due to R. Datko: Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3(1972), 428–445. Our approach is based on the extension of some techniques used in the deterministic case for the study of asymptotic behavior of skew-evolution semiflows in Banach spaces.
DOI : 10.1007/s10587-012-0071-0
Classification : 37L55, 60H25, 93E15
Keywords: stochastic skew-evolution semiflow; nonuniform exponential dichotomy in mean square 
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Stoica, Diana; Megan, Mihail. On nonuniform dichotomy for stochastic skew-evolution semiflows in Hilbert spaces. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 879-887. doi: 10.1007/s10587-012-0071-0

[1] Arnold, L.: Stochastic Differential Equations: Theory and Applications. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons (1974). | MR | Zbl

[2] Ateiwi, A. M.: About bounded solutions of linear stochastic Ito systems. Miskolc Math. Notes 3 (2002), 3-12 | DOI | MR

[3] Bensoussan, A., Flandoli, F.: Stochastic inertial manifold. Stochastics and Stochastics Reports 53 (1995), 13-39. | DOI | MR | Zbl

[4] Buse, C., Barbu, D.: The Lyapunov equations and nonuniform exponential stability. Stud. Cerc. Mat. 49 (1997), 25-31. | MR | Zbl

[5] Caraballo, T., Duan, J., Lu, K., Schmalfuss, B.: Invariant manifolds for random and stochastic partial differential equations. Adv. Nonlinear Stud. 10 (2010), 23-52. | DOI | MR | Zbl

[6] Prato, G. Da, Ichikawa, A.: Lyapunov equations for time-varying linear systems. Systems Control Lett. 9 (1987), 165-172. | DOI | MR | Zbl

[7] Prato, G. Da, Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications. 44 Cambridge etc. Cambridge University Press (1992). | MR | Zbl

[8] Datko, R.: Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. Anal. 3 (1972), 428-445. | DOI | MR | Zbl

[9] Flandoli, F.: Stochastic flows for nonlinear second-order parabolic SPDE. Ann. Probab. 24 (1996), 547-558. | DOI | MR | Zbl

[10] Lemle, L. D., Wu, L.: Uniqueness of $C_{0}$-semigroups on a general locally convex vector space and an application. Semigroup Forum 82 (2011), 485-496. | DOI | MR | Zbl

[11] Lupa., N., Megan, M., Popa, I. L.: On weak exponential stability of evolution operators in Banach spaces. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73 (2010), 2445-2450. | DOI | MR | Zbl

[12] Megan, M., Sasu, A. L., Sasu, B.: Nonuniform exponential unstability of evolution operators in Banach spaces. Glas. Mat., III. Ser. 36 (2001), 287-295. | MR | Zbl

[13] Mohammed, S. - E. A., Zhang, T., Zhao, H.: The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Mem. Am. Math. Soc. 196 (2008), 1-105. | MR | Zbl

[14] Skorohod, A. V.: Random Linear Operators, Transl. from the Russian. Mathematics and Its Applications. Soviet Series., D. Reidel Publishing Company, Dordrecht, Boston, Lancaster (1984). | MR

[15] Stoica, C., Megan, M.: Nonuniform behaviors for skew-evolution semiflows in Banach spaces. Operator theory live. Proceedings of the 22nd international conference on operator theory, Timişoara, Romania, July 3-8, 2008. Bucharest: The Theta Foundation. Theta Series in Advanced Mathematics 12 (2010), 203-211. | MR

[16] Stoica, D.: Uniform exponential dichotomy of stochastic cocycles. Stochastic Process. Appl. 12 (2010), 1920-1928. | DOI | MR | Zbl

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