Almost automorphic solution for some stochastic evolution equation driven by Lévy noise with coefficients S2−almost automorphic

Mamadou Moustapha Mbaye

Nonautonomous Dynamical Systems (2016)

  • Volume: 3, Issue: 1, page 85-103
  • ISSN: 2353-0626

Abstract

top
In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case when the functions forcing are both continuous and S2−almost automorphic. We provide an example to illustrate ours results.

How to cite

top

Mamadou Moustapha Mbaye. "Almost automorphic solution for some stochastic evolution equation driven by Lévy noise with coefficients S2−almost automorphic." Nonautonomous Dynamical Systems 3.1 (2016): 85-103. <http://eudml.org/doc/285879>.

@article{MamadouMoustaphaMbaye2016,
abstract = {In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case when the functions forcing are both continuous and S2−almost automorphic. We provide an example to illustrate ours results.},
author = {Mamadou Moustapha Mbaye},
journal = {Nonautonomous Dynamical Systems},
keywords = {almost automorphic solution; composition theorem; stochastic processes; stochastic evolution equations; Lévy noise; stochastic evolution equations},
language = {eng},
number = {1},
pages = {85-103},
title = {Almost automorphic solution for some stochastic evolution equation driven by Lévy noise with coefficients S2−almost automorphic},
url = {http://eudml.org/doc/285879},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Mamadou Moustapha Mbaye
TI - Almost automorphic solution for some stochastic evolution equation driven by Lévy noise with coefficients S2−almost automorphic
JO - Nonautonomous Dynamical Systems
PY - 2016
VL - 3
IS - 1
SP - 85
EP - 103
AB - In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case when the functions forcing are both continuous and S2−almost automorphic. We provide an example to illustrate ours results.
LA - eng
KW - almost automorphic solution; composition theorem; stochastic processes; stochastic evolution equations; Lévy noise; stochastic evolution equations
UR - http://eudml.org/doc/285879
ER -

References

top
  1. [1] D. Applebaum, Lévy Process and Stochastic Calculus, second edition, Cambridge University Press, (2009).  
  2. [2] J. Blot, G.M.Mophou, G.M. N’Guérékata and D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Analysis, Theory, Methods and Applications, 71, (3–4), (2009), 903–909.  
  3. [3] P. Bezandry and T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal, 86, (2007), 819–827. [Crossref] Zbl1130.34033
  4. [4] P. Bezandry, Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, Statist. Probab. Lett. 78, (2008), 2844–2849. [WoS] Zbl1156.60046
  5. [5] P. H. Bezandry and T. Diagana, Square-mean almost periodic solutions nonautonomous stochastic differential equations, Electron. J. Differential Equations, 2007, No. 117, 10 pp.  Zbl1138.60323
  6. [6] P. Bezandry and T. Diagana, Existence of S2-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. Theory Differ. Equ. 35, (2008), 1–19.  Zbl1183.34080
  7. [7] S.Bochner, Uniform convergence of monotone sequences of functions, Proc. Natl. Acad. Sci. USA, 47, (1961), 582-585. [Crossref] Zbl0103.05304
  8. [8] P. Cieutat, S. Fatajou and G.M. N’Guérékata, Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis 89, (1), (2010), 11–27. [WoS][Crossref] Zbl1186.43008
  9. [9] Y.K. Changa, Z. H. Zhaoa, G.M. N’Guérékata and R. Mab, Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Analysis: Real World Applications 12, (2011), 1130–1139.  
  10. [10] Z. Chen andW. Lin Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, Journal of Functional Analysis, 261, (2011), 69–89.  Zbl1233.60030
  11. [11] T. Diagana and M. M.Mbaye, Existence results for some nonlinear hyperbolic partial differential equations, Electron. J. Differ. Equ., Vol. 2015 (2015), No. 241, 1-10.  Zbl1328.43006
  12. [12] T. Diagana and M. M. Mbaye, Square-mean almost periodic solutions to some singular stochastic differential equations, Applied Mathematics Letters, 54, (2016), 48–53. [Crossref] Zbl1337.34062
  13. [13] M. A. Diop, K. Ezzinbi and M. M. Mbaye, Measure theory and S2− pseudo almost periodic and automorphic process: Application to stochastic evolution equations, Afrika Matematika, 26, (5), (2015), 779-812.  Zbl1326.34097
  14. [14] M. A. Diop, K. Ezzinbi and M. M. Mbaye, Existence and global attractiveness of a pseudo almost periodic solution in the p-th mean sense for stochastic evolution equation driven by a fractional Brownian, Stochastics: An International Journal Of Probability And Stochastic Processes, 87, (6), (2015), 1061-1093.  Zbl1337.60137
  15. [15] M. A. Diop, K. Ezzinbi and M. M. Mbaye, Measure theory and square-mean pseudo almost periodic and automorphic process: Application to stochastic evolution equations, Bulletin of the Malaysian Mathematical Sciences Society, DOI: 10.1007/s40840-015-0278-y. [Crossref] Zbl1326.34097
  16. [16] K. Ezzinbi and G.M. N’Guérékata, Almost automorphic solutions for some partial functional differential equations, Journal of Mathematical Analysis and Applications 328, (1), (2007), 344–358.  Zbl1121.34081
  17. [17] K. Ezzinbi and G.M. N’Guérékata, Almost automorphic solutions for partial functional differential equations with infinite delay, Semigroup Forum 75, (1), (2007), 95–115. [Crossref][WoS] Zbl1132.34059
  18. [18] K. Ezzinbi, V. Nelson and G.M. N’Guérékata, C(n)-almost automorphic solutions of some nonautonomous differential equations, Cubo 10, (2), (2008), 61–74.  Zbl1168.47033
  19. [19] M. M. Fu, Almost automorphic solutions for nonautonomous stochastic differential equations, Journal ofMathematical Analysis and Applications. 393, (2012), 231–238.  Zbl1244.60056
  20. [20] M.M. Fu and Z.X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc. 133, (2010), 3689-3701. [WoS] Zbl1202.60109
  21. [21] J.A. Goldstein and G.M. N’Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 133, (2005), 2401-2408. [WoS] Zbl1073.34073
  22. [22] M. Kamenskii, O. Mellah, P. Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl. 427 (2015) 336–364  Zbl06535673
  23. [23] O. Mellah, P. Raynaud de Fitte, Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients, Electron. J. Differential Equations 2013 (91) (2013) 1–7.  
  24. [24] J. Liang, G.M. N’Guérékata, T-J. Xiao and J. Zhang, Some properties of pseudo-almost automorphic functions and applications to abstract differential equations, Nonlinear Analysis, Theory, Methods and Applications 70, (7), (2009), 2731–2735.  Zbl1162.44002
  25. [25] J. Liang, J. Zhang and T-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, Journal of Mathematical Analysis and Applications 340, (2), (2008), 1493–1499.  Zbl1134.43001
  26. [26] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266, (3), (2014), 1115–1149. [WoS] Zbl1291.60121
  27. [27] G.M. N’Guérékata, Almost automorphic solutions to second-order semilinear evolution equations, Nonlinear Analysis, Theory, Methods and Applications 71, (12), (2009), e432–e435.  
  28. [28] G.M. N’Guérékata, Topics in almost Automorphy, Springer-verlag, New York, 2005.  
  29. [29] G.M. N’Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. TMA 68, (2008), 2658–2667.  Zbl1140.34399
  30. [30] S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, (2007).  Zbl1205.60122
  31. [31] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic evolution equations, Stoch. Anal. Appl. 13, (1995), 13–33. [Crossref] Zbl0816.60062
  32. [32] G.D. Prato and J. Zabczyk, Stochastics Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, (1992).  
  33. [33] C. Tudor, Almost periodic solutions of affine stochastic evolutions equations, Stoch. Stoch. Rep, 38, (1992), 251–266. [Crossref] Zbl0752.60049
  34. [34] Y. Wang, Z.X. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25, (2012), 2803–2821. [Crossref] Zbl1260.60114
  35. [35] T-J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76, (3), (2008), 518–524. [WoS][Crossref] Zbl1154.46023
  36. [36] T-J. Xiao, X-X. Zhu and J. Liang, Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Analysis, Theory, Methods and Applications, 70, (11), (2009), 4079–4085.  Zbl1175.34076
  37. [37] S. Zaidman, Almost automorphic solutions of some abstract evolutions equations, Istituto Lombardo. Accademia di Scienze e Lettere, Estrato dai Rendiconti, Classe di Scienze (A), 110, (1976), 578–588.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.