# Extinction time of logistic branching processes in a Brownian environment

## TL;DR

This paper investigates the extinction times of logistic branching processes influenced by a Brownian environment, using a Lamperti-type representation to connect these processes with Feller diffusions perturbed by spectrally positive Lévy processes, and characterizes their distributional properties.

## Contribution

It introduces a novel Lamperti-type representation linking logistic branching processes in a Brownian environment to Feller diffusions with Lévy perturbations, and derives explicit distributional results for extinction times.

## Key findings

- Processes converge to a specific distribution when perturbed by a subordinator.
- Extinction occurs almost surely without subordinator perturbation.
- Provides Laplace transform and expectation of absorption time via Ricatti equations.

## Abstract

In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive L\'evy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment converges to a specified distribution; otherwise, it becomes extinct a.s. In the latter scenario, and following a similar approach to Lambert (Lambert, Ann. Appl. Probab., 2005), we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Ricatti differential equation. In particular, the latter characterises the law of the process coming down from infinity.