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

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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

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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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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Keywords

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

Mathematics Subject Classification (2000)

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