Spectral analysis of stable processes on the positive half-line

TL;DR

This paper derives an explicit spectral expansion for the semigroup of a stable process killed on exiting the positive half-line, using Wiener-Hopf factorization and special functions, providing a novel example for non-symmetric Levy processes.

Contribution

It introduces the first explicit spectral expansion for a non-symmetric Levy process killed on the positive half-line, utilizing double sine functions and generalized Fourier transforms.

Findings

Explicit spectral expansion derived for stable processes

Eigenfunctions expressed via double sine functions

Generalized Fourier sine transform established

Abstract

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Levy process killed on the first exit form the positive half-line.