Spectral theory for one-dimensional symmetric Levy processes killed upon hitting the origin

TL;DR

This paper develops spectral theory for one-dimensional symmetric Levy processes killed at the origin, providing eigenfunction expansions and integral formulas for transition densities and hitting times.

Contribution

It introduces a new integral formula for eigenfunctions and eigenfunction expansion of transition operators for these processes under mild assumptions.

Findings

Eigenfunction expansion of the transition semigroup

Integral formulas for transition density

Distribution of hitting times derived

Abstract

Spectral theory for the transition semigroup of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is given. As an application, integral fomulae for the transition density and the distribution of the hitting time of the origin are proved.