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

DeBlassie, R. Dante; Méndez-Hernández, Pedro J.

doi:10.1090/s0002-9947-06-04032-3

Cited by 17 publications

(18 citation statements)

References 21 publications

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“…The important case of α = 1 was studied in detail by several authors, see [1,13] and the references therein. It is known that (λ n ) 1/α is continuous and increasing in α ∈ (0, 2], see [7,8,9,10]. For a discussion of related results and historical remarks, see e.g.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Eigenvalues of the fractional Laplace operator in the interval

Kwaśnicki

2012

Journal of Functional Analysis

143

237

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Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (−∆) α/2 (α ∈ (0, 2)) in the interval (−1, 1) is given: the n-th eigenvalue is equal to (nπ/2 − (2 − α)π/8) α + O(1/n). Simplicity of eigenvalues is proved for α ∈ [1, 2). L 2 and L ∞ properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Eigenvalues of the fractional Laplace operator in the interval

Kwaśnicki

2012

Journal of Functional Analysis

143

237

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“…The key problem in our paper is the description of eigenfunctions and eigenvalues of A D and P D t . The study of the spectral theoretic properties of the semigroups of killed symmetric α-stable processes has been the subject of many papers in recent years, see for example [1,2,3,14,15,16,18,19]. Our paper is a continuation of the work of Bañuelos and Kulczycki [1].…”

Section: Introductionmentioning

confidence: 88%

“…Similar or related methods were also applied e.g. by DeBlassie and Mendez-Hérnandez [17,18,19,33], and the idea can be traced back to the work of Spitzer [42], see also [34].…”

Section: Introductionmentioning

confidence: 99%

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Spectral properties of the Cauchy process on half-line and interval

Kulczycki

Kwaśnicki

Małecki

et al. 2010

Proceedings of the London Mathematical Society

134

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We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0, ∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A = − − d 2 dx 2 with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions ψ λ of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the half-line (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψ λ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λ n of A in the interval the asymptotic formula λ n = nπ 2 − π 8 + O( 1 n ) is derived, and all eigenvalues λ n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λ n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.

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“…At the top of page 531, we made the statement that This was pointed out to us by the anonymous referee of the article [2]. In fact, the conclusion of Theorem 1.5 is false, as one can show via a one-dimensional counterexample.…”

Section: Introductionmentioning

confidence: 95%

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

Cited by 17 publications

References 21 publications

Eigenvalues of the fractional Laplace operator in the interval

Eigenvalues of the fractional Laplace operator in the interval

Spectral properties of the Cauchy process on half-line and interval

Correction to “Higher Order PDEs and Symmetric Stable Processes,” Probability Theory and Related Fields 129, 495–536 (2004)

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