×

A class of multidimensional \(Q\)-processes. (English) Zbl 1133.60345

Summary: 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.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60J75 Jump processes (MSC2010)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, W. J. (1991). Continuous-Time Markov Chains . Springer, New York. · Zbl 0731.60067
[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, 42–52. JSTOR: · Zbl 0551.92013
[3] Brockwell, P. J. (1986). The extinction time of a general birth and death process with catastrophes. J. Appl. Prob. 23, 851–858. JSTOR: · Zbl 0614.60081
[4] Brockwell, P. J., Gani, J. and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731. JSTOR: · Zbl 0496.92007
[5] Cairns, B. and Pollett, P. K. (2004). Extinction times for a general birth, death and catastrophe process. J. Appl. Prob. 41, 1211–1218. · Zbl 1063.60106
[6] Chen, J. W. (1995). Positive recurrence of a finite-dimensional Brusselator model. Acta Math. Sci. 15, 121–125 (in Chinese). · Zbl 0900.92161
[7] Chen, M.-F. (1986). Coupling for jump processes. Acta Math. Sin. New Ser. 2, 123–136. · Zbl 0615.60078
[8] Chen, M.-F. (1992). From Markov Chains to Non-Equilibrium Particle Systems . World Scientific, Singapore. · Zbl 1078.60003
[9] Chen, M.-F. (1994). Optimal Markovian couplings and applications. Acta Math. Sin. New Ser. 10, 260–275. · Zbl 0813.60068
[10] Chen, M.-F. (2001). Explicit criteria for several types of ergodicity. Chinese J. Appl. Statist. 17, 113–120. · Zbl 1153.60385
[11] Haken, H. (1983). Synergetics: An Introduction , 3rd edn. Springer, Berlin. · Zbl 0523.93001
[12] Han, D. (1991). Ergodicity for one-dimensional Brusselator model. J. Xingjiang Univ. 8, 37–40 (in Chinese). · Zbl 0964.80502
[13] Mao, Y.-H. and Zhang, Y.-H. (2004). Exponential ergodicity for single-birth processes. J. Appl. Prob. 41, 1022–1032. · Zbl 1062.60089
[14] Nicolis, G. and Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems . John Wiley, New York. · Zbl 0363.93005
[15] Pakes, A. G. (1986). The Markov branching-catastrophe process. Stoch. Process. Appl. 23, 1–33. · Zbl 0633.92014
[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. 421–430. · Zbl 0114.09001
[17] Schlögl, F. (1972). Chemical reaction models for phase transitions. Z. Phys. 253, 147–161.
[18] Shao, J.-H. (2003). Estimates of eigenvalue for random walks on trees. Masters Thesis, 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, 575–577 (in Chinese). · Zbl 1090.92512
[20] Wu, B. and Zhang, Y.-H. (2005). One dimensional Brusselator model. Chinese J. Appl. Prob. Statist. 21, 225–234 (in Chinese). · Zbl 1167.60351
[21] Yan, S.-J. and Chen, M.-F. (1986). Multidimensional \(Q\)-processes. Chinese Ann. Math. 7B, 90–110. · Zbl 0596.60074
[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, 139–152 (in Chinese).
[23] Zhang, H.-J., Lin, X. and Hou, Z.-T. (2000). Uniformly polynomial convergence for standard transition functions. Chinese Ann. Math. 21A, 351–356 (in Chinese). · Zbl 0966.60032
[24] Zhang, Y.-H. (1994). The conservativity of coupling jump processes. J. Beijing Normal Univ. 30, 305–307 (in Chinese). · Zbl 0816.60081
[25] Zhang, Y.-H. (1996). Construction of order-preserving coupling for one-dimensional Markov chains. Chinese J. Appl. Prob. Statist. 12, 376–382 (in Chinese). · Zbl 0953.60532
[26] Zhang, Y.-H. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270–277. · Zbl 0984.60083
[27] Zhang, Y.-H. (2003). Moments of the first hitting time for single birth processes. J. Beijing Normal Univ. 39, 430–434 (in Chinese). · Zbl 1073.60073
[28] Zhang, Y.-H. (2004). The hitting time and stationary distribution for single birth processes. J. Beijing Normal Univ. 40, 157–161 (in Chinese). · Zbl 1075.60572
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.