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On the Shape of the Ground State Eigenfunction for Stable Processes

Authors:	Rodrigo Bañuelos Tadeusz Kulczycki Pedro J Méndez-Hernández

Institution:	(1) Mathematics Department, Purdue University, West Lafayette, IN, 47907;(2) Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław, Poland;(3) Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt lake City, UT, 84112-0090

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 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 ℝ^d, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination. Mathematics Subject Classification (2000) 30C45. Rodrigo Ba?uelos: R. Ba?uelos 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.

Keywords:	symmetric stable processes ground-state eigenfunctions multiple integrals
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