# American Institute of Mathematical Sciences

2016, 21(10): 3551-3573. doi: 10.3934/dcdsb.2016110

## Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes

 1 School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China 2 School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Received  January 2016 Revised  June 2016 Published  November 2016

In this paper, we are concerned with a class of neutral stochastic partial differential equations driven by $\alpha$-stable processes. By combining some stochastic analysis techniques, tools from semigroup theory and delay integral inequalities, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the $p$-th moment to the stochastic systems are obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to the stochastic systems under consideration. Last, an example is presented to illustrate our theory in the work.
Citation: Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus,, 2nd Edition. Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. [2] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching,, Statist. Probab. Lett., 79 (2009), 1663. doi: 10.1016/j.spl.2009.04.006. [3] J. Bao and C. Yuan, Numerical analysis for neutral SPDEs driven by $\alpha$-stable processes,, Infinite Dimen. Anal. Quant. Probab. Relat. Topics., 17 (2014). doi: 10.1142/S0219025714500313. [4] Z. Dong, L. Xu and X. C. Zhang, Invariant measures of stochastic 2D Navier-Stokes equation driven $\alpha$-stable processes,, Elec. Comm. Probab., 16 (2011), 678. doi: 10.1214/ECP.v16-1664. [5] U. Haagerup, The best constants in the Khintchine inequality,, Studia Math., 70 (1981), 231. [6] N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes,, North-Holland, (1981). [7] Y. Liu and J. L. Zhai, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise,, C. R. Acad. Sci. Paris, 350 (2012), 97. doi: 10.1016/j.crma.2011.11.017. [8] S. Long, L. Teng and D. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses,, Statist. Probab. Lett., 82 (2012), 1699. doi: 10.1016/j.spl.2012.05.018. [9] S. Mohammed, Stochastic Functional Differential Equation,, Pitman, (1984). [10] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [11] E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes,, Probab. Theory Relat. Fields., 149 (2011), 97. doi: 10.1007/s00440-009-0243-5. [12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance,, Chapman & Hall, (1994). [13] K. Sato, Lévy Processes and Infinitely Divisible Distributions,, Cambridge University Press, (1999). [14] F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes,, Stoch. Proc. Appl., 121 (2011), 466. doi: 10.1016/j.spa.2010.12.002. [15] J. Y. Wang and Y. L. Rao, A note on stability of SPDEs driven by $\alpha$-stable noises,, Adv. Difference Equations., 2014 (2014). doi: 10.1186/1687-1847-2014-98. [16] L. L. Wang and X. C. Zhang, Harnack inequalities for SDEs driven by cylindrical $\alpha$-stable processes,, Potential Anal., 42 (2015), 657. doi: 10.1007/s11118-014-9451-4. [17] L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noise,, Stoch. Proc. Appl., 123 (2013), 3710. doi: 10.1016/j.spa.2013.05.002. [18] D. Y. Xu and S. J. Long, Attracting and quasi-invariant sets of non-autonomous neural networks with delays,, Neurocomputing, 77 (2012), 222. doi: 10.1016/j.neucom.2011.09.004. [19] L. G. Xu and D. Y. Xu, $P$-attracting and $p$-invariant sets for a class of impulsive stochastic functional differential equations,, Comput. Math. Appl., 57 (2009), 54. doi: 10.1016/j.camwa.2008.09.027. [20] Y. C. Zang and J. P. Li, Stability in distribution of neutral stochastic partial differential delay equations driven by $\alpha$-stable process,, Adv. Difference Equations, 13 (2014). [21] X. C. Zhang, Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes,, Stoch. Proc. Appl., 123 (2013), 1213. doi: 10.1016/j.spa.2012.11.012. [22] Z. H. Zhao and J. G. Jian, Attracting and quasi-invariant sets for BAM neural networks of neutral-type with time-varying and infinite distributed delays,, Neurocomputing., 140 (2014), 265. doi: 10.1016/j.neucom.2014.03.015.

show all references

##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus,, 2nd Edition. Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. [2] J. Bao, Z. Hou and C. Yuan, Stability in distribution of neutral stochastic differential delay equations with Markovian switching,, Statist. Probab. Lett., 79 (2009), 1663. doi: 10.1016/j.spl.2009.04.006. [3] J. Bao and C. Yuan, Numerical analysis for neutral SPDEs driven by $\alpha$-stable processes,, Infinite Dimen. Anal. Quant. Probab. Relat. Topics., 17 (2014). doi: 10.1142/S0219025714500313. [4] Z. Dong, L. Xu and X. C. Zhang, Invariant measures of stochastic 2D Navier-Stokes equation driven $\alpha$-stable processes,, Elec. Comm. Probab., 16 (2011), 678. doi: 10.1214/ECP.v16-1664. [5] U. Haagerup, The best constants in the Khintchine inequality,, Studia Math., 70 (1981), 231. [6] N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes,, North-Holland, (1981). [7] Y. Liu and J. L. Zhai, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise,, C. R. Acad. Sci. Paris, 350 (2012), 97. doi: 10.1016/j.crma.2011.11.017. [8] S. Long, L. Teng and D. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses,, Statist. Probab. Lett., 82 (2012), 1699. doi: 10.1016/j.spl.2012.05.018. [9] S. Mohammed, Stochastic Functional Differential Equation,, Pitman, (1984). [10] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [11] E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processes,, Probab. Theory Relat. Fields., 149 (2011), 97. doi: 10.1007/s00440-009-0243-5. [12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance,, Chapman & Hall, (1994). [13] K. Sato, Lévy Processes and Infinitely Divisible Distributions,, Cambridge University Press, (1999). [14] F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes,, Stoch. Proc. Appl., 121 (2011), 466. doi: 10.1016/j.spa.2010.12.002. [15] J. Y. Wang and Y. L. Rao, A note on stability of SPDEs driven by $\alpha$-stable noises,, Adv. Difference Equations., 2014 (2014). doi: 10.1186/1687-1847-2014-98. [16] L. L. Wang and X. C. Zhang, Harnack inequalities for SDEs driven by cylindrical $\alpha$-stable processes,, Potential Anal., 42 (2015), 657. doi: 10.1007/s11118-014-9451-4. [17] L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noise,, Stoch. Proc. Appl., 123 (2013), 3710. doi: 10.1016/j.spa.2013.05.002. [18] D. Y. Xu and S. J. Long, Attracting and quasi-invariant sets of non-autonomous neural networks with delays,, Neurocomputing, 77 (2012), 222. doi: 10.1016/j.neucom.2011.09.004. [19] L. G. Xu and D. Y. Xu, $P$-attracting and $p$-invariant sets for a class of impulsive stochastic functional differential equations,, Comput. Math. Appl., 57 (2009), 54. doi: 10.1016/j.camwa.2008.09.027. [20] Y. C. Zang and J. P. Li, Stability in distribution of neutral stochastic partial differential delay equations driven by $\alpha$-stable process,, Adv. Difference Equations, 13 (2014). [21] X. C. Zhang, Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes,, Stoch. Proc. Appl., 123 (2013), 1213. doi: 10.1016/j.spa.2012.11.012. [22] Z. H. Zhao and J. G. Jian, Attracting and quasi-invariant sets for BAM neural networks of neutral-type with time-varying and infinite distributed delays,, Neurocomputing., 140 (2014), 265. doi: 10.1016/j.neucom.2014.03.015.
 [1] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [2] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [3] Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001 [4] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [5] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [6] Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic & Related Models, 2012, 5 (2) : 417-440. doi: 10.3934/krm.2012.5.417 [7] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [8] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [9] Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036 [10] Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979 [11] Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773 [12] Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601 [13] Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53 [14] Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339 [15] M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495 [16] Xiangjun Wang, Jianghui Wen, Jianping Li, Jinqiao Duan. Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1575-1584. doi: 10.3934/dcdsb.2012.17.1575 [17] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067 [18] Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025 [19] Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991 [20] Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002

2017 Impact Factor: 0.972