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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[5]

U. Haagerup, The best constants in the Khintchine inequality, Studia Math., 70 (1981), 231-283.  Google Scholar

[6]

N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Mohammed, Stochastic Functional Differential Equation, Pitman, Boston, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.  Google Scholar

[13]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Edition. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

U. Haagerup, The best constants in the Khintchine inequality, Studia Math., 70 (1981), 231-283.  Google Scholar

[6]

N. Ikeda and S. Watanable, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Mohammed, Stochastic Functional Differential Equation, Pitman, Boston, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.  Google Scholar

[13]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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