# Local times for spectrally negative L\'evy processes

## TL;DR

This paper derives explicit Laplace transforms involving local times for spectrally negative Lévy processes, connecting them with scale functions, permanental processes, and loop soup measures.

## Contribution

It introduces a novel approach to compute joint Laplace transforms of local times at various stopping times for spectrally negative Lévy processes, linking them to scale functions and other stochastic processes.

## Key findings

- Explicit Laplace transforms involving local times are obtained.
- Connections established with permanental processes and Markovian loop soups.
- Results enhance understanding of local times in spectrally negative Lévy processes.

## Abstract

For spectrally negative L\'evy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure.