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Two-sided heat kernel estimates for censored stable-like processes
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  • Published: 22 January 2009

Two-sided heat kernel estimates for censored stable-like processes

  • Zhen-Qing Chen1,
  • Panki Kim2 &
  • Renming Song3 

Probability Theory and Related Fields volume 146, pages 361–399 (2010)Cite this article

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Abstract

In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C 1,1 open sets in \({\mathbb R^d}\) , where d ≥ 1 and \({\alpha\in (1, 2)}\) . We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

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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 and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, South Korea

    Panki Kim

  3. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Renming Song

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

Correspondence to Panki Kim.

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Cite this article

Chen, ZQ., Kim, P. & Song, R. Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields 146, 361–399 (2010). https://doi.org/10.1007/s00440-008-0193-3

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  • Received: 03 April 2008

  • Revised: 08 November 2008

  • Published: 22 January 2009

  • Issue Date: March 2010

  • DOI: https://doi.org/10.1007/s00440-008-0193-3

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Keywords

  • Fractional Laplacian
  • Censored stable process
  • Censored stable-like process
  • Symmetric α-stable process
  • Symmetric stable-like process
  • Heat kernel
  • Transition density
  • Transition density function
  • Green function
  • Exit time
  • Lévy system
  • Boundary Harnack principle
  • Parabolic Harnack principle
  • Intrinsic ultracontractivity

Mathematics Subject Classification (2000)

  • Primary: 60J35
  • 47G20
  • 60J75
  • Secondary: 47D07
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