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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.

60H10Stochastic ordinary differential equations
Full Text: DOI
[1] Bochner, S.: A new approach to almost automorphicity, Proc. natl. Acad. sci. USA 48, 2039-2043 (1962) · Zbl 0112.31401 · doi:10.1073/pnas.48.12.2039
[2] Bochner, S.: Continuous mappings of almost automorphic and almost periodic functions, Proc. natl. Acad. sci. USA 52, 907-910 (1964) · Zbl 0134.30102 · doi:10.1073/pnas.52.4.907
[3] N’guérékata, G. M.: Almost automorphic and almost periodic functions in abstract space, (2001) · Zbl 1001.43001
[4] N’guérékata, G. M.: Topics in almost automorphy, (2005) · 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 · doi:10.1006/jmaa.1997.5899
[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 · doi:10.1023/A:1006266900851
[7] Abbas, S.; Bahuguna, D.: Almost periodic solutions of neutral functional differential equations, Comput. math. Appl. 55, 2593-2601 (2008) · Zbl 1142.34367 · doi:10.1016/j.camwa.2007.10.011
[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 · doi:10.1016/j.nonrwa.2009.10.024
[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 · doi:10.1016/j.na.2009.09.028
[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 · doi:10.1007/s00233-003-0021-0
[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 · doi:10.1016/j.aml.2006.05.015
[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 · doi:10.1080/00036810701355018
[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 · doi:10.1016/j.na.2007.02.012
[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 · doi:10.1016/j.na.2007.07.053
[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 · doi:10.1080/00036810701397788
[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 · doi:10.1016/j.spl.2008.04.008
[19] Bezandry, P.; Diagana, T.: Existence of S2-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. theory differ. Equ. 35, 1-19 (2008) · Zbl 1183.34080 · emis:journals/EJQTDE/2008/200835.html
[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) · 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 · doi:10.1080/07362999508809380
[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) · Zbl 0761.60052
[26] Ichikawa, A.: Stability of semilinear stochastic evolution equations, J. math. Anal. appl. 90, 12-44 (1982) · Zbl 0497.93055 · doi:10.1016/0022-247X(82)90041-5