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One-point reflection

Author

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  • Chen, Zhen-Qing
  • Fukushima, Masatoshi

Abstract

We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh’s Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh’s Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).

Suggested Citation

  • Chen, Zhen-Qing & Fukushima, Masatoshi, 2015. "One-point reflection," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1368-1393.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1368-1393
    DOI: 10.1016/j.spa.2014.11.002
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    References listed on IDEAS

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    1. Fukushima, Masatoshi, 2010. "From one dimensional diffusions to symmetric Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 590-604, May.
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    Cited by:

    1. Noba, Kei & Yano, Kouji, 2019. "Generalized refracted Lévy process and its application to exit problem," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1697-1725.
    2. Li, Liping & Li, Xiaodan, 2020. "Dirichlet forms and polymer models based on stable processes," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5940-5972.
    3. Chen, Zhen-Qing & Fukushima, Masatoshi, 2018. "Stochastic Komatu–Loewner evolutions and BMD domain constant," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 545-594.

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