Spectral analysis of subordinate Brownian motions in half-line

Abstract

We study one-dimensional Levy processes with Levy-Khintchine exponent psi(xi^2), where psi is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Levy measure has completely monotone density; or, equivalently, symmetric Levy processes whose Levy measure has completely monotone density on the positive half-line. Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the half-line. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.

58 pages, 1 figure. Major revision

References in corpus (9)

• Fractional Laplacian in Bounded Domains
• Spectral properties of the Cauchy process
• Two-sided Green function estimates for killed subordinate Brownian motions
• Suprema of Lévy processes
• Spectral analysis of subordinate Brownian motions in half-line
• The Spectrum of the Fractional Laplacian and First Passage Time Statistics
• First passage times for subordinate Brownian motions
• Potential Theory of Subordinate Brownian Motions Revisited
• Meromorphic Levy processes and their fluctuation identities

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• Spectral analysis of subordinate Brownian motions in half-line
• First passage times for subordinate Brownian motions
• One-dimensional quasi-relativistic particle in the box
• Null controllability from the exterior of a one-dimensional nonlocal heat equation
• Hitting times of points for symmetric Lévy processes with completely monotone jumps
• Spectral theory for one-dimensional symmetric Levy processes killed upon hitting the origin
• Rogers functions and fluctuation theory
• On Sobolev norms involving Hardy operators in a half-space
• Fine scales of decay of operator semigroups
• Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval
• Two-term spectral asymptotics for the Dirichlet pseudo-relativistic kinetic energy operator on a bounded domain