Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin

TL;DR

This paper derives integral formulas for the distribution of the first hitting time and the transition operators of one-dimensional alpha-stable processes killed at the origin, involving generalized eigenfunctions.

Contribution

It provides new spectral-type integral formulas for these processes, extending understanding of their behavior upon hitting the origin.

Findings

Integral formula for first hitting time distribution

Spectral integral formula for killed process transition operators

Use of generalized eigenfunctions with oscillating functions

Abstract

We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional $\alpha$-stable processes $X_t$, where $\alpha\in(1,2)$. We also find a spectral-type integral formula for the transition operators $P_0^t$ of $X_t$ killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for $P_0^t$.