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Stability of Martin boundary under non-local Feynman-Kac perturbations
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  • Published: 02 February 2004

Stability of Martin boundary under non-local Feynman-Kac perturbations

  • Zhen-Qing Chen1 &
  • Panki Kim1 

Probability Theory and Related Fields volume 128, pages 525–564 (2004)Cite this article

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Abstract.

Recently the authors showed that the Martin boundary and the minimal Martin boundary for a censored (or resurrected) α-stable process Y in a bounded C 1,1-open set D with α∈(1,2) can all be identified with the Euclidean boundary ∂D of D. Under the gaugeability assumption, we show that the Martin boundary and the minimal Martin boundary for the Schrödinger operator obtained from Y through a non-local Feynman-Kac transform can all be identified with ∂D. In other words, the Martin boundary and the minimal Martin boundary are stable under non-local Feynman-Kac perturbations. Moreover, an integral representation of nonnegative excessive functions for the Schrödinger operator is explicitly given. These results in fact hold for a large class of strong Markov processes, as are illustrated in the last section of this paper. As an application, the Martin boundary for censored relativistic stable processes in bounded C 1,1-smooth open sets is studied in detail.

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Authors and Affiliations

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

    Zhen-Qing Chen & Panki Kim

Authors
  1. Zhen-Qing Chen
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  2. Panki Kim
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Corresponding author

Correspondence to Zhen-Qing Chen.

Additional information

This research is supported in part by NSF Grant DMS-0071486 and a RRF Grant from University of Washington

Mathematics Subject Classification (2000): Primary 31C35, 60J45, 35J10; Secondary 60J50, 60J57

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Chen, ZQ., Kim, P. Stability of Martin boundary under non-local Feynman-Kac perturbations. Probab. Theory Relat. Fields 128, 525–564 (2004). https://doi.org/10.1007/s00440-003-0317-8

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  • Received: 09 July 2003

  • Revised: 20 October 2003

  • Published: 02 February 2004

  • Issue Date: April 2004

  • DOI: https://doi.org/10.1007/s00440-003-0317-8

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Keywords

  •  Green function
  • Martin kernel
  • Martin boundary
  • Feynman-Kac transform
  • Schrödinger semigroup
  • Non-local perturbation
  • Minimal harmonic function
  • Excessive function
  • Martin integral representation
  • Stable process
  • Resurrection
  • h-transform
  • Conditional Markov process
  • Conditional gauge
  • Kato class
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