We study one-dimensional Lévy processes with Lévy-Khintchine exponent ψ(ξ²), where ψ 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,∞). 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-3-2,
author = {Mateusz Kwa\'snicki},
title = {Spectral analysis of subordinate Brownian motions on the half-line},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {211-271},
zbl = {1241.60023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-3-2}
}
Mateusz Kwaśnicki. Spectral analysis of subordinate Brownian motions on the half-line. Studia Mathematica, Tome 204 (2011) pp. 211-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-3-2/