Spectral analysis of subordinate Brownian motions on the half-line
Volume 206 / 2011
Abstract
We study one-dimensional Lévy processes with Lévy–Khintchine exponent $\psi (\xi ^2)$, where $\psi $ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on $(0, \infty )$. 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.
