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Asymptotic Properties of a Mean-Field Model with a Continuous-State-Dependent Switching Process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Fubao Xi  and
G. Yin
Fubao Xi*
Affiliation:
Beijing Institute of Technology
G. Yin*
Affiliation:
Wayne State University
*
∗Postal address: Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China. Email address: xifb@bit.edu.cn
∗∗Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: gyin@math.wayne.edu
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Abstract

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This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

Keywords

Mean-field modelcontinuous-state-dependent switchingMarkovian switchingFeller continuitystrong Fellercontinuityexponential ergodicitystrong ergodicity

MSC classification

Secondary: 60J60: Diffusion processes 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 221 - 243
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported in part by the National Natural Science Foundation of China under grant number 10671037.

Supported in part by the National Science Foundation under grant DMS-0603287 and in part by the National Security Agency under grant MSPF-068-029.

References

[1] Basak, G. K., Bisi, A. and Ghosh, M. K. (1996). Stability of a random diffusion with linear drift. J. Math. Anal. Appl. 202, 604–622.Google Scholar
[2] Chen, M.-F. (2004). Eigenvalues, Inequalities, and Ergodic Theory. Springer, London.Google Scholar
[3] Chen, M.-F. (2004). From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, River Edge, NJ.Google Scholar
[4] Chen, M.-F. and Li, S.-F. (1989). Coupling methods for multidimensional diffusion processes. Ann. Prob. 17, 151–177.Google Scholar
[5] Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer, New York.Google Scholar
[6] Dawson, D. A. (1983). Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31, 29–85.Google Scholar
[7] Dawson, D. A. and Zheng, X. (1991). Law of large numbers and central limit theorem for unbounded Jump mean-field models. Adv. Appl. Math. 12, 293–326.Google Scholar
[8] Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23, 1671–1691.Google Scholar
[9] Hitsuda, M. and Mitoma, I. (1986). Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J. Multivariate Anal. 19, 311–328.Google Scholar
[10] Hu, G. (1994). Stochastic Forces and Nonlinear Systems. Shanghai Scientific and Technological Education Publishing House, Shanghai (in Chinese).Google Scholar
[11] Ichihara, K. and Kunita, H. (1974). A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitsth. 30, 235–254. (Correction: 39 (1977), 81–84.)Google Scholar
[12] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
[13] Khas'minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Stijhoff and Noordhoff, Alphen.Google Scholar
[14] Khas'minskiĭ, R. Z., Zhu, C. and Yin, G. (2007). Stability of regime-switching diffusions. Stoch. Process. Appl. 117, 1037–1051.Google Scholar
[15] Kliemann, W. (1987). Recurrence and invariant measures for degenerate diffusions. Ann. Prob. 15, 690–707.Google Scholar
[16] Mao, X. (1999). Stability of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 79, 45–67.Google Scholar
[17] Mao, Y.-H. (2002). Strong ergodicity for Markov processes by coupling methods. J. Appl. Prob. 39, 839–852.Google Scholar
[18] Mao, Y.-H. (2006). Convergence rates in strong ergodicity for Markov processes. Stoch. Process. Appl. 116, 1964–1964.Google Scholar
[19] Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Prob. 24, 542–574.Google Scholar
[20] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
[21] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.Google Scholar
[22] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548.Google Scholar
[23] Shiga, T. and Tanaka, H. (1985). Central limit theorem for a system of Markovian particles with mean field interactions. Z. Wahrscheinlichkeitsth. 69, 439–459.Google Scholar
[24] Skorokhod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. American Mathemathical Society, Providence, RI.Google Scholar
[25] Xi, F.-B. (2002). Stability for a random evolution equation with Gaussian perturbation. J. Math. Anal. Appl. 272, 458–472.Google Scholar
[26] Xi, F.-B. (2004). Stability of a random diffusion with nonlinear drift. Statist. Prob. Lett. 68, 273–286.Google Scholar
[27] Xi, F.-B. (2008). Feller property and exponential ergodicity of diffusion processes with state-dependent switching. Sci. China Ser. A 51, 329–342.Google Scholar
[28] Xi, F.-B. and Zhao, L.-Q. (2006). On the stability of diffusion processes with state-dependent switching. Sci. China Ser. A 49, 1258–1274.Google Scholar
[29] Yin, G. G. and Zhang, Q. (1998). Continuous-Time Markov Chains and Applications (Appl. Math. 37). Springer, New York.Google Scholar
[30] Yuan, C. and Lygeros, J. (2005). On the exponential stability of switching diffusion processes. IEEE Trans. Automatic Control 50, 1422–1426.Google Scholar
[31] Yuan, C. and Mao, X. (2003). Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 103, 277–291.Google Scholar
[32] Zhu, C. and Yin, G. (2007). Asymtotic properties of hybrid diffusion systems. SIAM J. Control Optimization 46, 1155–1179.Google Scholar