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One-point extensions of Markov processes by darning
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  • Published: 06 July 2007

One-point extensions of Markov processes by darning

  • Zhen-Qing Chen1 &
  • Masatoshi Fukushima2 

Probability Theory and Related Fields volume 141, pages 61–112 (2008)Cite this article

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Abstract

This paper is a continuation of the works by Fukushima–Tanaka (Ann Inst Henri Poincaré Probab Stat 41: 419–459, 2005) and Chen–Fukushima–Ying (Stochastic Analysis and Application, p.153–196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L 2-infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L 2-infinitesimal generator of the extended process. An important role in our investigation is played by the α-order approaching probability u α .

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Zhen-Qing Chen

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

    Masatoshi Fukushima

Authors
  1. Zhen-Qing Chen
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  2. Masatoshi Fukushima
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Corresponding author

Correspondence to Zhen-Qing Chen.

Additional information

The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206.

The research of M. Fukushima is supported in part by Grant-in-Aid for Scientific Research of MEXT No.19540125.

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Cite this article

Chen, ZQ., Fukushima, M. One-point extensions of Markov processes by darning. Probab. Theory Relat. Fields 141, 61–112 (2008). https://doi.org/10.1007/s00440-007-0080-3

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  • Received: 30 July 2006

  • Revised: 10 May 2007

  • Published: 06 July 2007

  • Issue Date: May 2008

  • DOI: https://doi.org/10.1007/s00440-007-0080-3

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Mathematics Subject Classification (2000)

  • 60J50
  • 60J25
  • 60J35
  • 31C25
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