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Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

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Abstract

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − Δ)α/2 with m ∈ (0, 1]) in half-space-like C 1, 1 open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1, 1 open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets established recently in Chen et al. (Stoch Process their Appl, 2011).

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Authors and Affiliations

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

    Zhen-Qing Chen

  2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul, 151-747, Republic of 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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Correspondence to Zhen-Qing Chen.

Additional information

Research of Zhen-Qing Chen was partially supported by NSF Grants DMS-0906743 and DMR-1035196.

The work of Panki Kim was supported by Mid-career Researcher Program through NRF grant funded by the MEST (2010-0027491).

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Chen, ZQ., Kim, P. & Song, R. Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets. Potential Anal 36, 235–261 (2012). https://doi.org/10.1007/s11118-011-9228-y

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  • DOI: https://doi.org/10.1007/s11118-011-9228-y

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