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A Class of Multidimensional Q-Processes

  • Bo Wu (a1) and Yu-Hui Zhang (a1)

Abstract

In this paper we present some necessary conditions for the uniqueness, recurrence, and ergodicity of a class of multidimensional Q-processes, using the dual Yan-Chen comparison method. Then the coupling method is used to study the multidimensional processes in a specific space. As applications, three models of particle systems are illustrated.

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Copyright

Corresponding author

Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
∗∗ Email address: zhangyh@bnu.edu.cn

References

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[1] Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.
[2] Brockwell, P. J. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 4252.
[3] Brockwell, P. J. (1986). The extinction time of a general birth and death process with catastrophes. J. Appl. Prob. 23, 851858.
[4] Brockwell, P. J., Gani, J. and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709731.
[5] Cairns, B. and Pollett, P. K. (2004). Extinction times for a general birth, death and catastrophe process. J. Appl. Prob. 41, 12111218.
[6] Chen, J. W. (1995). Positive recurrence of a finite-dimensional Brusselator model. Acta Math. Sci. 15, 121125 (in Chinese).
[7] Chen, M.-F. (1986). Coupling for Jump processes. Acta Math. Sin. New Ser. 2, 123136.
[8] Chen, M.-F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.
[9] Chen, M.-F. (1994). Optimal Markovian couplings and applications. Acta Math. Sin. New Ser. 10, 260275.
[10] Chen, M.-F. (2001). Explicit criteria for several types of ergodicity. Chinese J. Appl. Statist. 17, 113120.
[11] Haken, H. (1983). Synergetics: An Introduction, 3rd edn. Springer, Berlin.
[12] Han, D. (1991). Ergodicity for one-dimensional Brusselator model. J. Xingjiang Univ. 8, 3740 (in Chinese).
[13] Mao, Y.-H. and Zhang, Y.-H. (2004). Exponential ergodicity for single-birth processes. J. Appl. Prob. 41, 10221032.
[14] Nicolis, G. and Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems. John Wiley, New York.
[15] Pakes, A. G. (1986). The Markov branching-catastrophe process. Stoch. Process. Appl. 23, 133.
[16] Reuter, G. E. H. (1961). Competition processes. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. 2, University of California Press, Berkeley, CA, pp. 421430.
[17] Schlögl, F. (1972). Chemical reaction models for phase transitions. Z. Phys. 253, 147161.
[18] Shao, J.-H. (2003). Estimates of eigenvalue for random walks on trees. , Beijing Normal University (in Chinese).
[19] Wu, B. and Zhang, Y.-H. (2004). A property of one-dimensional Brusselator model. J. Beijing Normal Univ. 41, 575577 (in Chinese).
[20] Wu, B. and Zhang, Y.-H. (2005). One dimensional Brusselator model. Chinese J. Appl. Prob. Statist. 21, 225234 (in Chinese).
[21] Yan, S.-J. and Chen, M.-F. (1986). Multidimensional Q-processes. Chinese Ann. Math. 7B, 90110.
[22] Yan, S.-J. and Li, Z.-B. (1980). The stochastic models for non-equilibrium systems and formulation of master equations. Acta Phys. Sin. 29, 139152 (in Chinese).
[23] Zhang, H.-J., Lin, X. and Hou, Z.-T. (2000). Uniformly polynomial convergence for standard transition functions. Chinese Ann. Math. 21A, 351356 (in Chinese).
[24] Zhang, Y.-H. (1994). The conservativity of coupling Jump processes. J. Beijing Normal Univ. 30, 305307 (in Chinese).
[25] Zhang, Y.-H. (1996). Construction of order-preserving coupling for one-dimensional Markov chains. Chinese J. Appl. Prob. Statist. 12, 376382 (in Chinese).
[26] Zhang, Y.-H. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270277.
[27] Zhang, Y.-H. (2003). Moments of the first hitting time for single birth processes. J. Beijing Normal Univ. 39, 430434 (in Chinese).
[28] Zhang, Y.-H. (2004). The hitting time and stationary distribution for single birth processes. J. Beijing Normal Univ. 40, 157161 (in Chinese).

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A Class of Multidimensional Q-Processes

  • Bo Wu (a1) and Yu-Hui Zhang (a1)

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