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Heat kernel for non-local operators with variable order

Author

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  • Chen, Xin
  • Chen, Zhen-Qing
  • Wang, Jian

Abstract

Let α(x) be a measurable function taking values in [α1,α2] for 0<α1⩽α2<2, and κ(x,z) be a positive measurable function that is symmetric in z and bounded between two positive constants. Under uniform Hölder continuous assumptions on α(x) and x↦κ(x,z), we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order Lf(x)=∫Rd(f(x+z)−f(x)−〈∇f(x),z〉1{|z|⩽1})κ(x,z)|z|d+α(x)dz. In particular, we show that the operator L generates a conservative Feller process on Rd having strong Feller property, which is usually assumed a priori in the literature to study analytic properties of L via probabilistic approaches. Our near-diagonal estimates and lower bound estimates of the heat kernel depend on the local behavior of index function α(x). When α(x)≡α∈(0,2), our results recover some results by Chen and Kumagai (2003) and Chen and Zhang (2016).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3574-3647
    DOI: 10.1016/j.spa.2019.10.004
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    References listed on IDEAS

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    1. Kim, Panki & Lee, Jaehun, 2019. "Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2130-2173.
    2. 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.
    3. 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.
    4. Sandrić, Nikola, 2013. "Long-time behavior of stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1276-1300.
    5. Bass, Richard F. & Tang, Huili, 2009. "The martingale problem for a class of stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1144-1167, April.
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