# The entrance law of the excursion measure of the reflected process for some classes of L\'evy processes

## TL;DR

This paper derives integral formulas for the Laplace transform of the entrance law of reflected excursions in symmetric Lévy processes, providing explicit density expressions and analyzing their properties for specific subclasses.

## Contribution

It introduces new integral formulas for the entrance law of reflected Lévy process excursions, especially for subordinate Brownian motions and stable processes, linking them to eigenfunctions.

## Key findings

- Explicit Laplace transform formulas for entrance laws
- Density expressions in terms of eigenfunctions for specific processes
- Asymptotic analysis of the density derivatives

## Abstract

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric L\'evy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.