In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813-857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1-13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we study the process conditioned on eventual extinction where three regimes for the asymptotic behaviour of the extinction probability appear. Finally, the process conditioned on non-extinction and the process with immigration are given.Key words and phrases: Continuous state branching processes in random environment, Brownian motion, explosion and extinction probabilities, exponential functional of Brownian motion, Q-process, supercritical process conditioned on eventual extinction, continuous state branching processes with immigration in random environment.MSC 2000 subject classifications: 60G17, 60G51, 60G80. * Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n.

Let ξ = (ξt, t ≥ 0) be a real-valued Lévy process and define its associated exponential functional as followsMotivated by applications to stochastic processes in random environment, we study the asymptotic behaviour ofwhere F = (F (x), x ≥ 0) is a function with polynomial decay at infinity and which is non increasing for large x. In particular, under some exponential moment conditions on ξ, we find five different regimes that depend on the shape of the Laplace exponent of ξ. Our proof relies on a discretization of the exponential functional It(ξ) and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our results to three questions associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for self-similar continuous state branching processes in a Lévy random environment. Secondly, we focus on the asymptotic behaviour of the mean population size in a model with competition or logistic growth which is affected by a Lévy random environment and finally, we study the tail behaviour of the maximum of a diffusion in a Lévy random environment.Key words and phrases: Lévy processes, exponential functional, continuous state branching processes in random environment, explosion and extinction probabilities, logistic process, diffusions in random environment.MSC 2000 subject classifications: 60G17, 60G51, 60G80.

Recently in [2], the notion of a multi-type continuous-state branching process (with immigration) having d-types was introduced as a solution to an d-dimensional vectorvalued SDE. Preceding that, work on affine processes, originally motivated by mathematical finance, in [10] also showed the existence of such processes. See also more recent contributions in this direction due to [17] and [8]. Older work on multi-type continuous-state branching processes is more sparse but includes [33] and [27], where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of [16] we explore their extinction properties and pose a number of open problems.

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