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Potential theory of truncated stable processes

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Abstract

For any \(\alpha \in (0, 2)\), a truncated symmetric α-stable process is a symmetric Lévy process in \(\mathbb{R}^{d}\) with a Lévy density given by \(c|x|^{-d-\alpha} 1_{\{|x| < 1\}}\) for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails.

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Authors and Affiliations

  1. Department of Mathematics, Seoul National University, Seoul, 151-742, South Korea

    Panki Kim

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Renming Song

Authors
  1. Panki Kim
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  2. Renming Song
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Corresponding author

Correspondence to Renming Song.

Additional information

The research of Panki Kim is supported by Research Settlement Fund for the new faculty of Seoul National University. The research of Renming Song is supported in part by a joint US-Croatia grant INT 0302167.

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Kim, P., Song, R. Potential theory of truncated stable processes. Math. Z. 256, 139–173 (2007). https://doi.org/10.1007/s00209-006-0063-6

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  • DOI: https://doi.org/10.1007/s00209-006-0063-6

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