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Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation

Authors: Zhen-Qing Chen, Panki Kim and Renming Song
Journal: Trans. Amer. Math. Soc. 367 (2015), 5237-5270
MSC (2010): Primary 60J35, 47G20, 60J75; Secondary 47D07, 47D08
Published electronically: November 10, 2014
MathSciNet review: 3335416
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Abstract: In this paper we show that Dirichlet heat kernel estimates for a class of (not necessarily symmetric) Markov processes are stable under non-local Feynman-Kac perturbations. This class of processes includes, among others, (reflected) symmetric stable-like processes in closed $ d$-sets in $ \mathbb{R}^d$, killed symmetric stable processes, censored stable processes in $ C^{1, 1}$ open sets, as well as stable processes with drifts in bounded $ C^{1, 1}$ open sets. These two-sided estimates are explicit involving distance functions to the boundary.

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Additional Information

Zhen-Qing Chen
Affiliation: Department of Mathematics and Research Institute of Mathematics, University of Washington, Seattle, Washington 98195

Panki Kim
Affiliation: Department of Mathematics, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu, Seoul 151-747, Republic of Korea

Renming Song
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Keywords: Fractional Laplacian, symmetric $\alpha$-stable process, symmetric stable-like process, censored stable process, relativistic symmetric stable process, heat kernel, transition density, Dirichlet heat kernel, Feynman-Kac perturbation, Feynman-Kac transform
Received by editor(s): May 3, 2012
Received by editor(s) in revised form: May 7, 2013, and May 17, 2013
Published electronically: November 10, 2014
Additional Notes: The research of Zhen-Qing Chen was partially supported by NSF Grants DMS-0906743, DMS-1206276 and DMR-1035196.
The research of Panki Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No.2009-0083521)
The research of Renming Song was supported in part by a grant from the Simons Foundation (208236)
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