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Oscillation of harmonic functions for subordinate Brownian motion and its applications

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  • Kim, Panki
  • Lee, Yunju

Abstract

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in a bounded κ-fat open set; if u is a positive harmonic function with respect to X in a bounded κ-fat open set D and h is a positive harmonic function in D vanishing on Dc, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h.

Suggested Citation

  • Kim, Panki & Lee, Yunju, 2013. "Oscillation of harmonic functions for subordinate Brownian motion and its applications," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 422-445.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:422-445
    DOI: 10.1016/j.spa.2012.09.015
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    1. 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. 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.

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