×

A topology for limits of Markov chains. (English) Zbl 1322.60153

Summary: In the investigation of limits of Markov chains, the presence of states which become instantaneous states in the limit may prevent the convergence of the chain in the Skorokhod topology. We present in this article a weaker topology adapted to handle this situation. We use this topology to derive the limit of random walks among random traps and sticky zero-range processes.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60F05 Central limit and other weak theorems
60F99 Limit theorems in probability theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beltrán, J.; Landim, C., Tunneling and metastability of continuous time Markov chains, J. Stat. Phys., 140, 1065-1114 (2010) · Zbl 1223.60061
[2] Beltrán, J.; Landim, C., Metastability of reversible finite state Markov processes, Stochastic Process. Appl., 121, 1633-1677 (2011) · Zbl 1223.60060
[3] Beltrán, J.; Landim, C., Metastability of reversible condensed zero range processes on a finite set, Probab. Theory Related Fields, 152, 781-807 (2012) · Zbl 1251.60070
[4] Beltrán, J.; Landim, C., Tunneling and metastability of continuous time Markov chains II, J. Stat. Phys., 149, 598-618 (2012) · Zbl 1260.82063
[5] Beltrán, J.; Landim, C., A Martingale approach to metastability, Probab. Theory Related Fields (2014), in press. http://dx/doi.org/10.1007/s00440-014-0549-9, arXiv:1305.5987 · Zbl 1338.60194
[6] Caputo, P.; Lacoin, H.; Martinelli, F.; Simenhaus, F.; Toninelli, F. L., Polymer dynamics in the depinned phase: metastability with logarithmic barriers, Probab. Theory Related Fields, 153, 587-641 (2012) · Zbl 1262.60093
[7] Caputo, P.; Martinelli, F.; Toninelli, F. L., On the approach to equilibrium for a polymer with adsorption and repulsion, Electron. J. Probab., 13, 213-258 (2008) · Zbl 1191.60110
[8] Chung, K. L., (Markov Chains with Stationary Transition Probabilities. Markov Chains with Stationary Transition Probabilities, Die Grundlehren der mathematischen Wissenschaften, Band 104 (1967), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. New York)
[9] Freedman, D., Markov Chains (1983), Springer-Verlag: Springer-Verlag New York, Berlin, Corrected reprint of the 1971 original · Zbl 0212.49801
[10] Jara, M.; Landim, C.; Teixeira, A., Quenched scaling limits of trap models, Ann. Probab., 39, 176-223 (2011) · Zbl 1211.60040
[11] Jara, M.; Landim, C.; Teixeira, A., Universality of trap models in the ergodic time scale, Ann. Probab., 42, 6, 2497-2557 (2014) · Zbl 1309.60098
[13] Landim, C., Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes, Comm. Math. Phys., 330, 1-32 (2014) · Zbl 1305.82045
[14] Lawler, G. F., Intersections of Random Walks, Probability and its Applications (1991), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 1228.60004
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.