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$ \alpha$-continuity properties of the symmetric $ \alpha$-stable process


Authors: R. Dante DeBlassie and Pedro J. Méndez-Hernández
Journal: Trans. Amer. Math. Soc. 359 (2007), 2343-2359
MSC (2000): Primary 60J45
DOI: https://doi.org/10.1090/S0002-9947-06-04032-3
Published electronically: December 19, 2006
MathSciNet review: 2276623
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Abstract: Let $ D$ be a domain of finite Lebesgue measure in $ \mathbb{R}^d$ and let $ X^D_t$ be the symmetric $ \alpha$-stable process killed upon exiting $ D$. Each element of the set $ \{ \lambda_i^\alpha\}_{i=1}^\infty$ of eigenvalues associated to $ X^D_t$, regarded as a function of $ \alpha\in(0,2)$, is right continuous. In addition, if $ D$ is Lipschitz and bounded, then each $ \lambda_i^\alpha$ is continuous in $ \alpha$ and the set of associated eigenfunctions is precompact.


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Additional Information

R. Dante DeBlassie
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Email: deblass@math.tamu.edu

Pedro J. Méndez-Hernández
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Address at time of publication: Escuela de Matemática, Universidad de Costa Rica, San Pedro de Montes de Oca, Costa Rica
Email: mendez@math.utah.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04032-3
Keywords: Symmetric $\alpha$-stable process, eigenvalues, eigenfunctions
Received by editor(s): July 9, 2004
Received by editor(s) in revised form: April 4, 2005
Published electronically: December 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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