Extinction time of logistic branching processes in a Brownian environment

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