Stochastic Liénard equations with state-dependent switching

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• 作者：Fu-bao Xi ; G. Yin
• 关键词：stochastic Liénard equation ; state ; dependent switching ; strong Feller property ; positive Harris recurrence ; exponential ergodicity ; 60J60 ; 60J27 ; 34D25
• 刊名：Acta Mathematicae Applicatae Sinica, English Series
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：31
• 期：4
• 页码：893-908
• 全文大小：300 KB
• 参考文献：[1]Basak, G.K., Bisi, A., Ghosh, M.K. Stability of a random diffusion with linear drift. J. Math. Anal. Appl., 202: 604-22 (1996)MATH MathSciNet CrossRef
  [2]Basak, G.K., Bisi, A., Ghosh, M.K. Stability of a degenerate diffusions with state-dependent switching. J. Math. Anal. Appl., 240: 219-48 (1999)MATH MathSciNet CrossRef
  [3]Chen, M.F. From Markov Chains to Non-equilibrium Particle Systems, Second Edition. World Scientific, Singapore, 2004MATH CrossRef
  [4]Chen, Z.Q., Zhao, Z. Potential theory for elliptic systems. Ann. Probab., 24: 293-19 (1996)MATH MathSciNet CrossRef
  [5]Down, D., Meyn, S.P., Tweedie, R.L. Exponential and uniform ergodicity of Markov processes. Ann. Probab., 23: 1671-691 (1995)MATH MathSciNet CrossRef
  [6]Ghosh, M.K., Arapostathis, A., Marcus, S.I. Ergodic control of switching diffusions. SIAM J. Contr. Optim., 35: 1952-988 (1997)MATH MathSciNet CrossRef
  [7]Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981MATH
  [8]Khasminskii, R.Z. Stochastic Stability of Differential Equatio8ns. Stijhoff and Noordhoff, Alphen, 1980CrossRef
  [9]Khasminskii, R.Z., Zhu, C., Yin, G. Stability of regime-switching diffusions. Stoch. Proc. Appl., 117: 1037-051 (2007)MATH MathSciNet CrossRef
  [10]Mao, X.R. Stability of stochastic differential equations with Markovian switching. Stoch. Proc. Appl., 79: 45-7 (1999)MATH CrossRef
  [11]Mao, X.R. Stochastic Differential Equations and Applications, Second Edition. Horwood, Chichester, UK, 2008CrossRef
  [12]Meyn, S.P., Tweedie, R.L. Markov Chains and Stochastic Stability. Springer-Verlag, Berlin, 1993MATH CrossRef
  [13]Meyn, S.P., Tweedie, R.L. Stability of Markovian processes I: Discrete time chains. Adv. Appl. Probab., 24: 542-74 (1992)MATH MathSciNet CrossRef
  [14]Meyn, S.P., Tweedie, R.L. Stability of Markovian processes II: Continuous time processes and sampled chains. Adv. Appl. Probab., 25: 487-17 (1993)MATH MathSciNet CrossRef
  [15]Meyn, S.P., Tweedie, R.L. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25: 518-48 (1993)MATH MathSciNet CrossRef
  [16]Narita, K. Asymptotic behavior of solutions of SDE for relaxation oscillations. SIAM J. Math. Anal., 24: 172-99 (1993)MATH MathSciNet CrossRef
  [17]?ksendal, B. Stochastic Differential Equations, Sixth Edition. Springer-Verlag, Berlin, 2007
  [18]Skorohod, A.V. Asymptotic Methods in the Theory of Stochastic Differential Equations. Amer. Math. Soc., Providence, 1989
  [19]Wu, L.M. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Proc. Appl., 91: 205-38 (2001)MATH CrossRef
  [20]Xi, F.B. Stability of a random diffusion with nonlinear drift. Stat. Probab. Letters, 68: 273-86 (2004)MATH CrossRef
  [21]Xi, F.B. Feller property and exponential ergodicity of diffusion processes with state-dependent switching. Sci. China, Ser. A, 51: 329-42 (2008)MATH MathSciNet CrossRef
  [22]Xi, F.B., Zhao, L.Q. On the stability of diffusion processes with state-dependent switching. Sci. China, Ser. A, 49: 1258-274 (2006)MATH MathSciNet CrossRef
  [23]Yan, J.A. Measure and Integration. Shanxi Normal University Press, Xi’an, 1988 (in Chinese)
  [24]Yin, G., Zhu, C. Hybrid Switching Diffusions: Properties and Applications. Springer-Verlag, New York, 2010CrossRef
  [25]Yuan, C.G., Mao, X.R. Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Proc. Appl., 103: 277-91 (2003)MATH MathSciNet CrossRef
  [26]Zhu, C., Yin, G. Asymtotic properties of hybrid diffusion systems. SIAM J. Contr. Optim., 46: 1155-179 (2007)MATH MathSciNet CrossRef
  
• 作者单位：Fu-bao Xi (1)
  G. Yin (3)
  
  1. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China
  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA
  
• 刊物主题：Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics;
• 出版者：Institute of Applied Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
• ISSN：1618-3932

文摘

This work focuses on stochastic Liénard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly. Keywords stochastic Liénard equation state-dependent switching strong Feller property positive Harris recurrence exponential ergodicity