α-稳定噪声驱动的随机Volterra-Levin方程解的稳定性
Stability of Solutions for Stochastic Volterra-Levin Equations Driven by α-Stable Noise
摘要: 本文研究了α-稳定噪声驱动的随机Volterra-Levin方程。在一定条件下,得到了该类方程的解部分过程的依分布稳定性。
Abstract: In this paper, we study stochastic Volterra-Levin equations driven by α-stable noise. We have a try to deal with the stability conditions in distribution of the segment process of the solutions to the stochastic systems.
文章引用:饶维亚, 蔺焕泉, 姜童. α-稳定噪声驱动的随机Volterra-Levin方程解的稳定性[J]. 理论数学, 2019, 9(10): 1187-1194. https://doi.org/10.12677/PM.2019.910145

参考文献

[1] Volterra, V. (1982) Sur la théorie mathématique des phénomes héréditaires. Journal de Mathématiques Pures et Appliquées, 7, 249-298.
[2] Levin, J.J. (1963) The Asymptotic Behavior of the Solution of a Volterra Equation. Proceedings of the American Mathematical Society, 14, 534-541. [Google Scholar] [CrossRef
[3] MacCamy, R.C. and Wong, J.S.W. (1972) Stability Theorems for Some Functional Equations. Transactions of the American Mathematical Society, 164, 1-37. [Google Scholar] [CrossRef
[4] Burton, T.A. (1979) Stability Theory for Volterra Equations. Journal of Differential Equations, 32, 101-118. [Google Scholar] [CrossRef
[5] Gushchin, A. and Küchler, U. (2000) On Stationary Solutions of Delay Differential Equations Driven by Lévy Process. Stochastic Processes and their Applications, 88, 195-211. [Google Scholar] [CrossRef
[6] Liu, K. (2010) Retarded Stationary Ornstein-Uhlenbeck Processes Driven by Lévy Noise and Operator Self-Decomposability. Potential Analysis, 33, 291-312. [Google Scholar] [CrossRef
[7] Reiβ, M., Riedleb, M. and van Gaansc, O. (2006) Delay Differ-ential Equations Driven by Lévy Processes: Stationarity and Feller Properties. Stochastic Processes and their Applications, 116, 1409-1432. [Google Scholar] [CrossRef
[8] Li, D.S. and Xu, D.Y. (2012) Existence and Global Attractivity of Periodic Solution for Impulsive Stochastic Volterra-Levin Equations. Electron. Qualitative Theory of Differential Equations, 46, 1-12.
[9] Appleby, J.A.D. (2008) Fixed Points, Stability and Harmless Stochastic Perturbations.
[10] Burton, T.A. (2006) Stability by Fixed Point Theory for Functional Differential Equations. Dover, New York.
[11] Luo, J. (2010) Fixed Points and Exponential Stability for Stochastic Volterra-Levin Equations. Journal of Computational and Applied Mathematics, 234, 934-940. [Google Scholar] [CrossRef
[12] Zhao, D., Yuan, S. and Zhang, T. (2014) Improved Stability Conditions for a Class of Stochastic Volterra-Levin Equations. Applied Mathematics and Computation, 231, 39-47. [Google Scholar] [CrossRef
[13] Guo, L. and Zhu, Q. (2011) Stability Analysis for Stochastic Volterra-Levin Equations with Poisson Jumps: Fixed Point Approach. Journal of Mathematical Physics, 52, Article ID: 042702. [Google Scholar] [CrossRef
[14] Yin, H., Xiao, S., Xiao, X. and Wen, X. (2014) pth Moment Stability in Stochastic Neutral Volterra-Levin Equation with Lévy Noise and Variable Delays. Advances in Difference Equations, 2014, 106. [Google Scholar] [CrossRef
[15] Basak, G.K., Bisi, A. and Ghosh, M.K. (1996) Stability of a Random Diffusion with Linear Drift. Journal of Mathematical Analysis and Applications, 202, 604-622. [Google Scholar] [CrossRef
[16] 鲍建海. 马尔可夫调制的中立型随机微分方程的数值解及依分布稳定性[D]: [硕士学位论文]. 长沙: 中南大学, 2006.
[17] Bao, J., Hou, Z. and Yuan, C. (2009) Stability in Distribution of Neutral Stochastic Differential Delay Equations with Markovian Switching. Statistics and Probability Letters, 79, 1663-1673. [Google Scholar] [CrossRef
[18] Bao, J., Hou, Z. and Yuan, C. (2010) Stability in Distribution of Mild Solutions to Stochastic Partial Differential Equations. Proceedings of the American Mathematical Society, 138, 2169-2180. [Google Scholar] [CrossRef
[19] Bao, J., Truman, A. and Yuan, C. (2009) Stability in Distribution of Mild Solutions to Stochastic Partial Differential Delay Equations with Jumps. Proceedings of the Royal Society of London. Series A, 465, 2111-2134. [Google Scholar] [CrossRef
[20] Hu, G.X. and Wang, K. (2012) Stability in Distribution of Neutral Stochastic Functional Differential Equations with Markovian Switching. Journal of Mathematical Analysis and Appli-cations, 385, 757-769. [Google Scholar] [CrossRef
[21] Yuan, C. and Mao, X. (2003) Asymptotic Stability in Distribution of Stochastic Differential Equations with Markovian Switching. Stochastic Processes and their Applications, 103, 277-291. [Google Scholar] [CrossRef
[22] Li, Z. and Zhang, W. (2017) Stability in Distribution of Stochastic Volterra-Levin Equations. Statistics and Probability Letters, 122, 20-27. [Google Scholar] [CrossRef
[23] Priola, E. and Zabczyk, J. (2011) Structural Properties of Semilinear SPDEs Driven by Cylindrical Stable Processes. Probability Theory and Related Fields, 149, 97-137. [Google Scholar] [CrossRef
[24] Zang, Y. and Li, J. (2014) Stability in Distribution of Neutral Stochastic Partial Differential Delay Equations Driven by α-Stable Process. Advances in Difference Equations, 2014, 13. [Google Scholar] [CrossRef
[25] Mohammed, S. (1984) Stochastic Functional Differential Equation. Pitman, Boston, MA.