Spectral gap for stable process on convex planar double symmetric domains

2006/10/09 by Bartlomiej Dyda, Dyda, Bartlomiej, Tadeusz Kulczycki +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP) #math.PR #math.SP

paper · pdf · doi:10.48550/arxiv.math/0610283

arxiv created 2006/10/09 · arxiv updated 2009/12/01

Abstract

We study the semigroup of the symmetric α-stable process in bounded domains in \R². We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using "midconcavity" of the first eigenfunction, we obtain sharp upper and lower bound estimates of the spectral gap.

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