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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 |
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. |
show all references
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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