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Potential theory of subordinate Brownian motions with Gaussian components

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

Abstract

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C1,1 open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:764-795
    DOI: 10.1016/j.spa.2012.11.007
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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.
    2. Kim, Panki & Song, Renming & Vondracek, Zoran, 2009. "Boundary Harnack principle for subordinate Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1601-1631, May.
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    Cited by:

    1. Chen, Zhen-Qing & Wang, Jie-Ming, 2022. "Boundary Harnack principle for diffusion with jumps," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 342-395.
    2. Fotopoulos, Stergios & Jandhyala, Venkata & Wang, Jun, 2015. "On the joint distribution of the supremum functional and its last occurrence for subordinated linear Brownian motion," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 149-156.
    3. 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.
    4. Liu, Rongli & Ren, Yan-Xia & Song, Renming, 2022. "Convergence rate for a class of supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 286-327.
    5. Ren, Yan-Xia & Song, Renming & Sun, Zhenyao, 2020. "Limit theorems for a class of critical superprocesses with stable branching," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4358-4391.
    6. 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.

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