Spectral analysis of subordinate Brownian motions on the half-line

Mateusz Kwaśnicki¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-976 Warszawa, Poland and Institute of Mathematics and Computer Science Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 206 (2011) no. 3, pp. 211-271

Résumé

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.

Détail

DOI : 10.4064/sm206-3-2

Keywords: study one dimensional processes khintchine exponent psi where psi complete bernstein function these processes subordinate brownian motions corresponding subordinators whose measure has completely monotone density equivalently symmetric processes whose measure has completely monotone density infty examples include symmetric stable processes relativistic processes main result formula generalized eigenfunctions transition operators process killed after exiting half line generalized eigenfunction expansion transition operators derived application formula distribution first passage time supremum functional obtained