# Extinction rate of continuous state branching processes in critical L\'evy environments

## TL;DR

This paper investigates the extinction speed of continuous state branching processes in oscillating Lévy environments, extending previous stable case results by analyzing path behavior conditioned on environmental infimum.

## Contribution

It introduces a novel path analysis approach for branching processes in Lévy environments under Spitzer's condition, generalizing prior stable environment findings.

## Key findings

- Derived extinction rate formulas under Lévy environment oscillation
- Extended stability-based results to more general Lévy processes
- Provided insights into process behavior conditioned on environmental infimum

## Abstract

We study the speed of extinction of continuous state branching processes in a L\'evy environment, where the associated L\'evy process oscillates. Assuming that the L\'evy process satisfies the Spitzer's condition and the existence of some exponential moments, we extend recent results where the associated branching mechanism was stable. Our study relies on the path analysis of the process together with its environment, when this latter is conditioned to have a non negative running infimum. This approach is inspired from the discrete setting with i.i.d. environment studied in (Afanasyev et al. 2005).