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On the Boundary Theory of Subordinate Killed Lévy Processes

  • Panki Kim
  • Renming SongEmail author
  • Zoran Vondraček
Article
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Abstract

Let Z be a subordinate Brownian motion in \(\mathbb {R}^{d}\), d ≥ 2, via a subordinator with Laplace exponent ϕ. We kill the process Z upon exiting a bounded open set \(D\subset \mathbb {R}^{d}\) to obtain the killed process ZD, and then we subordinate the process ZD by a subordinator with Laplace exponent ψ. The resulting process is denoted by YD. Both ϕ and ψ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of YD, in particular the boundary theory. First, in case that D is a κ-fat bounded open set, we show that the Harnack inequality holds. If, in addition, D satisfies the local exterior volume condition, then we prove the Carleson estimate. In case D is a smooth open set and the lower weak scaling index of ψ is strictly larger than 1/2, we establish the boundary Harnack principle with explicit decay rate near the boundary of D. On the other hand, when ψ(λ) = λγ with γ ∈ (0, 1/2], we show that the boundary Harnack principle near the boundary of D fails for any bounded C1,1 open set D. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of YD. Under an additional condition on ψ, we establish sharp two-sided estimates of the jumping kernel of YD which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set E strictly contained in D, showing that the behavior of YD in the interior of D is determined by the composition ψϕ.

Keywords

Lévy processes Subordination Green functions Jumping kernels Harmonic functions Harnack inequality Boundary Harnack principle Carleson estimate 

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Notes

Acknowledgments

We thank the referee for carefully reading the manuscript and providing some useful suggestions.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversityGwanak-gu SeoulRepublic of Korea
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.School of Mathematical SciencesNankai UniversityTianjinPeople’s Republic of China
  4. 4.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

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