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Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations. (English) Zbl 1209.60034
Summary: We introduce a new concept of Stepanov-like almost automorphy (or \(S^{2}\)-almost automorphy) for stochastic processes. We use the results obtained to investigate the existence and uniqueness of a Stepanov-like almost automorphic mild solution to a class of nonlinear stochastic differential equations in a real separable Hilbert space. Our main results extend some known ones in the sense of square-mean almost automorphy.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Bochner, S., A new approach to almost automorphicity, Proc. natl. acad. sci. USA, 48, 2039-2043, (1962) · Zbl 0112.31401
[2] Bochner, S., Continuous mappings of almost automorphic and almost periodic functions, Proc. natl. acad. sci. USA, 52, 907-910, (1964) · Zbl 0134.30102
[3] N’Guérékata, G.M., Almost automorphic and almost periodic functions in abstract space, (2001), Kluwer Academic Plenum Publishers New York, London, Moscow · Zbl 1001.43001
[4] N’Guérékata, G.M., Topics in almost automorphy, (2005), Springer New York, Boston, Dordrecht, London, Moscow · Zbl 1073.43004
[5] Hernández, E.; Henríquez, H.R., Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. math. anal. appl., 221, 499-522, (1998) · Zbl 0926.35151
[6] Henríquez, H.R.; Vasquez, C.H., Almost periodic solutions of abstract retarded functional differential equations with unbounded delay, Acta appl. math., 57, 105-132, (1999) · Zbl 0944.34058
[7] Abbas, S.; Bahuguna, D., Almost periodic solutions of neutral functional differential equations, Comput. math. appl., 55, 2593-2601, (2008) · Zbl 1142.34367
[8] Zhao, Z.H.; Chang, Y.K.; Li, W.S., Asymptotically almost periodic, almost periodic and pseudo almost periodic mild solutions for neutral differential equations, Nonlinear anal. RWA, 11, 3037-3044, (2010) · Zbl 1205.34088
[9] Zhao, Z.H.; Chang, Y.K.; Nieto, J.J., Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear anal. TMA, 72, 1886-1894, (2010) · Zbl 1189.34116
[10] N’Guérékata, G.M., Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup forum, 69, 80-86, (2004) · Zbl 1077.47058
[11] Diagana, T.; N’Guérékata, G.M., Almost automorphic solutions to semilinear evolution equations, Funct. differ. equ., 13, 195-206, (2006) · Zbl 1102.34044
[12] Diagana, T.; N’Guérékata, G.M., Almost automorphic solutions to some classes of partial evolution equations, Appl. math. lett., 20, 462-466, (2007) · Zbl 1169.35300
[13] Diagana, T.; N’Guérékata, G.M., Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. anal., 86, 723-733, (2007) · Zbl 1128.43006
[14] N’Guérékata, G.M.; Pankov, A., Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear anal., 68, 2658-2667, (2008) · Zbl 1140.34399
[15] Lee, H.; Alkahby, H., Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear anal., 69, 2158-2166, (2008) · Zbl 1162.34063
[16] Y.K. Chang, Z.H. Zhao, J.J. Nieto, Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. Mat. Complut. (2010), doi:10.1007/s13163-010-0047-2. · Zbl 1232.34087
[17] Bezandry, P.; Diagana, T., Existence of almost periodic solutions to some stochastic differential equations, Appl. anal., 86, 819-827, (2007) · Zbl 1130.34033
[18] Bezandry, P., Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statist. probab. lett., 78, 2844-2849, (2008) · Zbl 1156.60046
[19] Bezandry, P.; Diagana, T., Existence of \(S^2\)-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. qual. theory differ. equ., 35, 1-19, (2008) · Zbl 1183.34080
[20] Dorogovtsev, A.Ya.; Ortega, O.A., On the existence of periodic solutions of a stochastic equation in a Hilbert space, Visnik kiiv. univ. ser. mat. mekh., 30, 21-30, (1988), 115 · Zbl 0900.60072
[21] Da Prato, G.; Tudor, C., Periodic and almost periodic solutions for semilinear stochastic evolution equations, Stoch. anal. appl., 13, 13-33, (1995) · Zbl 0816.60062
[22] Tudor, C., Almost periodic solutions of affine stochastic evolutions equations, Stoch. stoch. rep., 38, 251-266, (1992) · Zbl 0752.60049
[23] Tudor, C.A.; Tudor, M., Pseudo almost periodic solutions of some stochastic differential equations, Math. rep. (bucur.), 1, 305-314, (1999) · Zbl 1019.60058
[24] M.M. Fu, Z.X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc. (in press), arXiv:1001.3049v1 [math.DS]. · Zbl 1202.60109
[25] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[26] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
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