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

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
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show all references

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2nd Edition. Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[2]

Statist. Probab. Lett., 79 (2009), 1663-1673. doi: 10.1016/j.spl.2009.04.006.  Google Scholar

[3]

Infinite Dimen. Anal. Quant. Probab. Relat. Topics., 17 (2014), 1450031, 16 pp. doi: 10.1142/S0219025714500313.  Google Scholar

[4]

Elec. Comm. Probab., 16 (2011), 678-688. doi: 10.1214/ECP.v16-1664.  Google Scholar

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Studia Math., 70 (1981), 231-283.  Google Scholar

[6]

North-Holland, Amsterdam, 1981.  Google Scholar

[7]

C. R. Acad. Sci. Paris, Ser. I., 350 (2012), 97-100. doi: 10.1016/j.crma.2011.11.017.  Google Scholar

[8]

Statist. Probab. Lett., 82 (2012), 1699-1709. doi: 10.1016/j.spl.2012.05.018.  Google Scholar

[9]

Pitman, Boston, 1984. Google Scholar

[10]

Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[11]

Probab. Theory Relat. Fields., 149 (2011), 97-137. doi: 10.1007/s00440-009-0243-5.  Google Scholar

[12]

Chapman & Hall, New York, 1994.  Google Scholar

[13]

Cambridge University Press, Cambridge, 1999.  Google Scholar

[14]

Stoch. Proc. Appl., 121 (2011), 466-478. doi: 10.1016/j.spa.2010.12.002.  Google Scholar

[15]

Adv. Difference Equations., 2014 (2014), 11pp. doi: 10.1186/1687-1847-2014-98.  Google Scholar

[16]

Potential Anal., 42 (2015), 657-669. doi: 10.1007/s11118-014-9451-4.  Google Scholar

[17]

Stoch. Proc. Appl., 123 (2013), 3710-3736. doi: 10.1016/j.spa.2013.05.002.  Google Scholar

[18]

Neurocomputing, 77 (2012), 222-228. doi: 10.1016/j.neucom.2011.09.004.  Google Scholar

[19]

Comput. Math. Appl., 57 (2009), 54-61. doi: 10.1016/j.camwa.2008.09.027.  Google Scholar

[20]

Adv. Difference Equations, 13 (2014), 16pp.  Google Scholar

[21]

Stoch. Proc. Appl., 123 (2013), 1213-1228. doi: 10.1016/j.spa.2012.11.012.  Google Scholar

[22]

Neurocomputing., 140 (2014), 265-272. doi: 10.1016/j.neucom.2014.03.015.  Google Scholar

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