Spectral properties of the Cauchy process on half‐line and interval
By Tadeusz Kulczycki, Mateusz Kwaśnicki, J. Małecki, A. Stos. 2009
76 citations recorded by the source corpus.
We study the spectral properties of the transition semigroup of the killed one‐dimensional 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=−−(d2/dx2) 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 ψλ of 𝒜 is derived, and then used to construct a spectral representation of 𝒜. Explicit formulas for the transition density of the killed Cauchy process on 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 ψλ is also used to construct approximations to eigenfunctions of 𝒜 in the interval. For the eigenvalues λn of 𝒜 in the interval the asymptotic formula λn = n π/2 − π/8 + O(1/n) is derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/27c883fc3f24befa9576c410387f4e78d79b5af6
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Citations in the corpus
- Ten equivalent definitions of the fractional laplace operator (2015, methodology): “It was heavily used, for example, in hydrodynamics [22, 23, 28, 31], and, more recently, in probability theory [1, 2, 8, 32].”
- Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, unknown): “4] bounds the survival probability of the relativistic process in a half-line, and [31] gives an explicit formula for the transition density of the killed Cauchy process (α = 1) on the half-line.”
- Eigenvalues of the fractional Laplace operator in the interval (2010, background): “Auxiliary estimates Define, as in [13], Appendix C, an auxiliary function:”