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Convergence of symmetric Markov chains on \({\mathbb{Z}^d}\)
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  • Published: 09 May 2009

Convergence of symmetric Markov chains on \({\mathbb{Z}^d}\)

  • Richard F. Bass1,
  • Takashi Kumagai2 &
  • Toshihiro Uemura3 

Probability Theory and Related Fields volume 148, pages 107–140 (2010)Cite this article

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  • 10 Citations

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Abstract

For each n let \({Y^{(n)}_t}\) be a continuous time symmetric Markov chain with state space \({n^{-1} \mathbb{Z}^d}\) . Conditions in terms of the conductances are given for the convergence of the \({Y^{(n)}_t}\) to a symmetric Markov process Y t on \({\mathbb{R}^d}\) . We have weak convergence of \(\{{Y^{(n)}_t: t \leq t_0\}}\) for every t 0 and every starting point. The limit process Y has a continuous part and may also have jumps.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Richard F. Bass

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Takashi Kumagai

  3. Department of Mathematics, Kansai University, Suita, Osaka, 564-8680, Japan

    Toshihiro Uemura

Authors
  1. Richard F. Bass
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  2. Takashi Kumagai
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  3. Toshihiro Uemura
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Corresponding author

Correspondence to Richard F. Bass.

Additional information

R. F. Bass’s research was partially supported by NSF grant DMS-0601783. T. Kumagai’s research was partially supported by the Grant-in-Aid for Scientific Research (B) 20340017 (Japan). T. Uemura’s research was partially supported by the Grant-in-Aid for Scientific Research (C) 20540130 (Japan).

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Cite this article

Bass, R.F., Kumagai, T. & Uemura, T. Convergence of symmetric Markov chains on \({\mathbb{Z}^d}\) . Probab. Theory Relat. Fields 148, 107–140 (2010). https://doi.org/10.1007/s00440-009-0224-8

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  • Received: 21 July 2008

  • Revised: 15 April 2009

  • Published: 09 May 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0224-8

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Keywords

  • Symmetric
  • Markov chains
  • Non-local
  • Dirichlet forms
  • Weak convergence
  • Elliptic diffusions
  • Central limit theorem

Mathematics Subject Classification (2000)

  • Primary: 60J10
  • Secondary: 60F05
  • 60J27
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