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

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 − (m2/α − )α/2 with m ∈ (0, 1]) in half-space-like C1,1 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 C1,1 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 C1,1 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 C1,1 open sets established recently in Chen et al. (Stoch Process their Appl, 2011). 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). Z.-Q. Chen (B) Department of Mathematics, University of Washington, Seattle, WA 98195, USA e-mail: zqchen@uw.edu P. Kim Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea e-mail: pkim@snu.ac.kr R. Song Department of Mathematics, University of Illinois, Urbana, IL 61801, USA e-mail: rsong@math.uiuc.edu 236 Z.-Q. Chen et al.

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Cite this paper

@inproceedings{Chen2012GlobalHK, title={Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets}, author={Zhen-Qing Chen and ·Panki Kim and ·Renming Song}, year={2012} }