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Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

Author

Listed:
  • Renming Song

    (University of Illinois)

  • Zoran Vondraček

    (University of Zagreb)

Abstract

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.

Suggested Citation

  • Renming Song & Zoran Vondraček, 2006. "Potential Theory of Special Subordinators and Subordinate Killed Stable Processes," Journal of Theoretical Probability, Springer, vol. 19(4), pages 817-847, December.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:4:d:10.1007_s10959-006-0045-y
    DOI: 10.1007/s10959-006-0045-y
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    Cited by:

    1. Hristo Sendov & Shen Shan, 2015. "New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1689-1725, December.
    2. Andreas E. Kyprianou & Víctor Rivero & Renming Song, 2010. "Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem," Journal of Theoretical Probability, Springer, vol. 23(2), pages 547-564, June.

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