Spectral analysis of subordinate Brownian motions in half-line

TL;DR

This paper derives formulas for eigenfunctions and transition operators of subordinate Brownian motions in a half-line, enabling explicit calculations of first passage time distributions for these processes.

Contribution

It provides a novel eigenfunction expansion for transition operators of subordinate Brownian motions killed at the boundary, facilitating analysis of their first passage times.

Findings

Derived explicit formulas for generalized eigenfunctions.

Established eigenfunction expansion for transition operators.

Obtained distribution formulas for first passage times.

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.