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Global uniform boundary Harnack principle with explicit decay rate and its application

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  • Kim, Panki
  • Song, Renming
  • Vondraček, Zoran

Abstract

In this paper, we consider a large class of subordinate Brownian motions X via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius R>0, we get a global uniform boundary Harnack principle with explicit decay rate. Our boundary Harnack principle is global in the sense that it holds for all R>0 and the comparison constant does not depend on R, and it is uniform in the sense that it holds for all balls with radii r≤R and the comparison constant depends neither on D nor on r. As an application, we give sharp two-sided estimates for the transition densities and Green functions of such subordinate Brownian motions in the half-space.

Suggested Citation

  • Kim, Panki & Song, Renming & Vondraček, Zoran, 2014. "Global uniform boundary Harnack principle with explicit decay rate and its application," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 235-267.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:235-267
    DOI: 10.1016/j.spa.2013.07.007
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    1. Kim, Panki & Song, Renming & Vondraček, Zoran, 2013. "Potential theory of subordinate Brownian motions with Gaussian components," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 764-795.
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    Cited by:

    1. Grzywny, Tomasz & Kim, Kyung-Youn & Kim, Panki, 2020. "Estimates of Dirichlet heat kernel for symmetric Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 431-470.
    2. Ante Mimica & Stjepan Šebek, 2019. "Harnack Inequality for Subordinate Random Walks," Journal of Theoretical Probability, Springer, vol. 32(2), pages 737-764, June.
    3. Vondraček, Zoran & Wagner, Vanja, 2018. "On purely discontinuous additive functionals of subordinate Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 707-725.
    4. Grzywny, Tomasz & Kwaśnicki, Mateusz, 2018. "Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 1-38.
    5. Kim, Kyung-Youn & Kim, Panki, 2014. "Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C1,η open sets," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3055-3083.
    6. Mimica, Ante & Vondraček, Zoran, 2014. "Unavoidable collections of balls for isotropic Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1303-1334.
    7. Kim, Panki & Song, Renming & Vondraček, Zoran, 2016. "Minimal thinness with respect to subordinate killed Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1226-1263.
    8. Xie, Longjie, 2017. "Singular SDEs with critical non-local and non-symmetric Lévy type generator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3792-3824.

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