We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.
. [J]. Frontiers of Mathematics in China, 2019, 14(2): 261-280.
Jing CUI, Wenping RONG. Existence and stability of μ-pseudo almost automorphic solutions for stochastic evolution equations. Front. Math. China, 2019, 14(2): 261-280.
P HBezandry. Existence of almost periodic solutions for semilinear stochastic evolution equations driven by fractional Brownian motion. Electron J Differential Equations, 2012, 2012: 1–21
2
JBlot, PCieutat, KEzzinbi. Measure theory and pseudo almost automorphic functions: New developments and applications. Nonlinear Anal, 2012, 75: 2426–2447 https://doi.org/10.1016/j.na.2011.10.041
3
JBlot, PCieutat, KEzzinbi. New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications. Appl Anal, 2013, 92: 493–526 https://doi.org/10.1080/00036811.2011.628941
4
SBochner. Continuous mappings of almost automorphic and almost periodic functions. Proc Natl Acad Sci USA, 1964, 52: 907–910 https://doi.org/10.1073/pnas.52.4.907
5
J FCao, Q GYang, Z THuang. Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations. Stochastics, 2011, 83: 259–275 https://doi.org/10.1080/17442508.2010.533375
6
Y KChang, Z HZhao, G MN'Guérékata. Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space. Adv Difference Equ, 2011, 2011: 1–12 https://doi.org/10.1186/1687-1847-2011-9
7
ZChen, WLin. Square-mean pseudo almost automorphic process and its application to stochastic evolution equations. J Funct Anal, 2011, 261: 69–89 https://doi.org/10.1016/j.jfa.2011.03.005
8
CCuevas, MPinto. Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain. Nonlinear Anal, 2001, 45: 73–83 https://doi.org/10.1016/S0362-546X(99)00330-2
9
GDa Prato, CTudor. Periodic and almost periodic solutions for semilinear stochastic equations. Stoch Anal Appl, 1995, 13: 13–33 https://doi.org/10.1080/07362999508809380
TDiagana, EHernández, MRabello. Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay. Math Comput Modelling, 2007, 45: 1241–1252 https://doi.org/10.1016/j.mcm.2006.10.006
12
H SDing, JLiang, G MN'Guérékata, T JXiao. Pseudo almost periodicity of some nonautonomous evolution equation with delay. Nonlinear Anal, 2007, 67: 1412–1218 https://doi.org/10.1016/j.na.2006.07.026
13
M ADiop, KEzzinbi, M MMbaye. Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion. Stochastics, 2015, 87: 1061–1093 https://doi.org/10.1080/17442508.2015.1026345
14
M ADiop, KEzzinbi, M MMbaye. Existence and global attractiveness of a square-mean μ-pseudo almost automorphic solution for some stochastic evolution equation driven by Lévy noise. Math Nachr, 2017, 290: 1260–1280 https://doi.org/10.1002/mana.201500345
15
M MFu, Z XLiu. Square-mean almost automorphic solutions for some stochastic differential equations. Proc Amer Math Soc, 2010, 138: 3689–3701 https://doi.org/10.1090/S0002-9939-10-10377-3
16
Z RHu, ZJin. Existence and uniqueness of pseudo almost automorphic mild solutions to some classes of partial hyperbolic evolution equations. Discrete Dyn Nat Soc, 2008, 2008: 405092 https://doi.org/10.1155/2008/405092
17
X LLi, Y LHan, B FLiu. Square-mean almost automorphic solutions to some stochastic evolution equations I: autonomous case. Acta Math Appl Sin Engl Ser, 2015, 31: 577–590 https://doi.org/10.1007/s10255-015-0487-z
18
JLiang, G MN'Guérékata, T JXiao, JZhang. Some properties of pseudo-almost automorphic functions and applications to abstract differential equations. Nonlinear Anal, 2009, 70: 2731–2735 https://doi.org/10.1016/j.na.2008.03.061
19
JLiang, JZhang, T JXiao. Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J Math Anal Appl, 2008, 340: 1493–1499 https://doi.org/10.1016/j.jmaa.2007.09.065
20
J HLiu, X QSong. Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations. J Funct Anal, 2010, 258: 196–207 https://doi.org/10.1016/j.jfa.2009.06.007
RSakthivel, PRevathi, S MAnthoni. Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations. Nonlinear Anal, 2012, 75: 3339–3347 https://doi.org/10.1016/j.na.2011.12.028
23
JSeidler, Prato-Zabczyk'sDa . maximal inequality revisited I. Math Bohem, 1993, 118: 67–106
24
T JXiao, JLiang, JZhang. Pseudo almost automorphic solutions to semilinear differential equations in Banach space. Semigroup Forum, 2008, 76: 518–524 https://doi.org/10.1007/s00233-007-9011-y
25
T JXiao, X XZhu, JLiang. Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal, 2009, 70: 4079–4085 https://doi.org/10.1016/j.na.2008.08.018
26
RYuan. Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlinear Anal, 2000, 41: 871–890 https://doi.org/10.1016/S0362-546X(98)00316-2
27
Q PZang. Asymptotic behaviour of the trajectory fitting estimator for reected Ornstein-Uhlenbeck processes. J Theoret Probab, 2019, 32: 183–201 https://doi.org/10.1007/s10959-017-0796-7
C YZhang, F LYang. Pseudo almost periodic solutions to parabolic boundary value inverse problems. Sci China Ser A, 2008, 7: 1203–1214 https://doi.org/10.1007/s11425-008-0006-2
30
Z YZhao, Y KChang, J JNieto. Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equations in Banach spaces. Nonlinear Anal, 2010, 72: 1886–1894 https://doi.org/10.1016/j.na.2009.09.028
