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Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive \(\alpha\)-stable processes. (English) Zbl 1354.60072
Summary: 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G52 Stable stochastic processes
60G15 Gaussian processes
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[1] D. Applebaum, <em>Lévy Processes and Stochastic Calculus</em>,, 2nd Edition. Cambridge University Press, (2009) · Zbl 1200.60001
[2] J. Bao, Stability in distribution of neutral stochastic differential delay equations with Markovian switching,, Statist. Probab. Lett., 79, 1663, (2009) · Zbl 1175.34103
[3] J. Bao, Numerical analysis for neutral SPDEs driven by \(α\)-stable processes,, Infinite Dimen. Anal. Quant. Probab. Relat. Topics., 17, (2014) · Zbl 1322.65013
[4] Z. Dong, Invariant measures of stochastic 2D Navier-Stokes equation driven \(α\)-stable processes,, Elec. Comm. Probab., 16, 678, (2011) · Zbl 1243.60052
[5] U. Haagerup, The best constants in the Khintchine inequality,, Studia Math., 70, 231, (1981) · Zbl 0501.46015
[6] N. Ikeda, <em>Stochastic Differential Equations and Diffusion Processes</em>,, North-Holland, (1981) · Zbl 0495.60005
[7] Y. Liu, A note on time regularity of generalized Ornstein-Uhlenbeck processes with cylindrical stable noise,, C. R. Acad. Sci. Paris, 350, 97, (2012) · Zbl 1280.60038
[8] S. Long, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses,, Statist. Probab. Lett., 82, 1699, (2012) · Zbl 1250.93124
[9] S. Mohammed, <em>Stochastic Functional Differential Equation</em>,, Pitman, (1984) · Zbl 0584.60066
[10] A. Pazy, <em>Semigroup of Linear Operators and Applications to Partial Differential Equations</em>,, Springer-Verlag, (1983) · Zbl 0516.47023
[11] E. Priola, Structural properties of semilinear SPDEs driven by cylindrical stable processes,, Probab. Theory Relat. Fields., 149, 97, (2011) · Zbl 1231.60061
[12] G. Samorodnitsky, <em>Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance,</em>, Chapman & Hall, (1994) · Zbl 0925.60027
[13] K. Sato, <em>Lévy Processes and Infinitely Divisible Distributions</em>,, Cambridge University Press, (1999) · Zbl 0973.60001
[14] F. Y. Wang, Gradient estimate for Ornstein-Uhlenbeck jump processes,, Stoch. Proc. Appl., 121, 466, (2011) · Zbl 1223.60069
[15] J. Y. Wang, A note on stability of SPDEs driven by \(α\)-stable noises,, Adv. Difference Equations., 2014, (2014) · Zbl 1343.60089
[16] L. L. Wang, Harnack inequalities for SDEs driven by cylindrical \(α\)-stable processes,, Potential Anal., 42, 657, (2015) · Zbl 1319.60137
[17] L. Xu, Ergodicity of the stochastic real Ginzburg-Landau equation driven by \(α\)-stable noise,, Stoch. Proc. Appl., 123, 3710, (2013) · Zbl 1295.60077
[18] D. Y. Xu, Attracting and quasi-invariant sets of non-autonomous neural networks with delays,, Neurocomputing, 77, 222, (2012)
[19] L. G. Xu, \(P\)-attracting and \(p\)-invariant sets for a class of impulsive stochastic functional differential equations,, Comput. Math. Appl., 57, 54, (2009) · Zbl 1165.60329
[20] Y. C. Zang, Stability in distribution of neutral stochastic partial differential delay equations driven by \(α\)-stable process,, Adv. Difference Equations, 13, (2014) · Zbl 1343.34178
[21] X. C. Zhang, Derivative formulas and gradient estimates for SDEs driven by \(α\)-stable processes,, Stoch. Proc. Appl., 123, 1213, (2013) · Zbl 1261.60060
[22] Z. H. Zhao, Attracting and quasi-invariant sets for BAM neural networks of neutral-type with time-varying and infinite distributed delays,, Neurocomputing., 140, 265, (2014)
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