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On the Shape of the Ground State Eigenfunction for Stable Processes

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Abstract

We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (−1,1) is concave on \((-\frac{1}{2},\frac{1}{2})\) . We call this property “mid-concavity”. A similar statement holds for rectangles in ℝd, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination.

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Authors and Affiliations

  1. Mathematics Department, Purdue University, West Lafayette, IN, 47907

    Rodrigo Bañuelos

  2. Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspianskiego 27, 50-370, Wrocław, Poland

    Tadeusz Kulczycki

  3. Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt lake City, UT, 84112-0090

    Pedro J. Méndez-Hernández

Authors
  1. Rodrigo Bañuelos
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  2. Tadeusz Kulczycki
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  3. Pedro J. Méndez-Hernández
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Corresponding author

Correspondence to Rodrigo Bañuelos.

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Mathematics Subject Classification (2000)

30C45.

Rodrigo Bañuelos: R. Bañuelos was supported in part by NSF grant # 9700585-DMS.

Tadeusz Kulczycki: T. Kulczycki was supported by KBN grant 2 P03A 041 22 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP.

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Bañuelos, R., Kulczycki, T. & Méndez-Hernández, P.J. On the Shape of the Ground State Eigenfunction for Stable Processes. Potential Anal 24, 205–221 (2006). https://doi.org/10.1007/s11118-005-8569-9

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  • DOI: https://doi.org/10.1007/s11118-005-8569-9

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