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

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December 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

Kai Liu ^1, and Zhi Li ^2,

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

Abstract
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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.

Keywords: Global attracting set, exponential decay in the $p$-th moment, $\alpha$-stable process., stability in distribution.

Mathematics Subject Classification: 60H15, 60G15, 39B0.

Citation: Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete and 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, Cambridge, 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), 1663-1673. 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), 1450031, 16 pp. 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), 678-688. doi: 10.1214/ECP.v16-1664.
[5]	U. Haagerup, The best constants in the Khintchine inequality, Studia Math., 70 (1981), 231-283.
[6]	N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 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, Ser. I., 350 (2012), 97-100. 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), 1699-1709. doi: 10.1016/j.spl.2012.05.018.
[9]	S. Mohammed, Stochastic Functional Differential Equation, Pitman, Boston, 1984.
[10]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 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), 97-137. 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, New York, 1994.
[13]	K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.
[14]	F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478. 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), 11pp. 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), 657-669. 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), 3710-3736. 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), 222-228. 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), 54-61. 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), 16pp.
[21]	X. C. Zhang, Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes, Stoch. Proc. Appl., 123 (2013), 1213-1228. 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), 265-272. doi: 10.1016/j.neucom.2014.03.015.

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References:

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition. Cambridge University Press, Cambridge, 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), 1663-1673. 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), 1450031, 16 pp. 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), 678-688. doi: 10.1214/ECP.v16-1664.
[5]	U. Haagerup, The best constants in the Khintchine inequality, Studia Math., 70 (1981), 231-283.
[6]	N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 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, Ser. I., 350 (2012), 97-100. 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), 1699-1709. doi: 10.1016/j.spl.2012.05.018.
[9]	S. Mohammed, Stochastic Functional Differential Equation, Pitman, Boston, 1984.
[10]	A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 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), 97-137. 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, New York, 1994.
[13]	K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.
[14]	F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478. 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), 11pp. 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), 657-669. 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), 3710-3736. 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), 222-228. 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), 54-61. 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), 16pp.
[21]	X. C. Zhang, Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes, Stoch. Proc. Appl., 123 (2013), 1213-1228. 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), 265-272. doi: 10.1016/j.neucom.2014.03.015.

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