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Heat kernel estimates for Δ+Δα/2 under gradient perturbation

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  • Chen, Zhen-Qing
  • Hu, Eryan

Abstract

For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a→0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.

Suggested Citation

  • Chen, Zhen-Qing & Hu, Eryan, 2015. "Heat kernel estimates for Δ+Δα/2 under gradient perturbation," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2603-2642.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2603-2642
    DOI: 10.1016/j.spa.2015.02.016
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    References listed on IDEAS

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    1. Chen, Zhen-Qing & Kumagai, Takashi, 2003. "Heat kernel estimates for stable-like processes on d-sets," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 27-62, November.
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    Cited by:

    1. Chen, Xin & Chen, Zhen-Qing & Wang, Jian, 2020. "Heat kernel for non-local operators with variable order," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3574-3647.
    2. Zhang, Bin & Yao, Zhigang & Liu, Junfeng, 2023. "On a class of mixed stochastic heat equations driven by spatially homogeneous Gaussian noise," Statistics & Probability Letters, Elsevier, vol. 196(C).

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