Spectral analysis of subordinate Brownian motions on the half-line

Volume 206 / 2011

Mateusz Kwaśnicki Studia Mathematica 206 (2011), 211-271 MSC: Primary 47G30, 60G51; Secondary 60G52, 60J35. DOI: 10.4064/sm206-3-2


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.


  • Mateusz KwaśnickiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-976 Warszawa, Poland
    Institute of Mathematics and Computer Science
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image