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.

No abstract

Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration (CBI process) converges almost surely.If an x log(x) moment condition on the branching mechanism does not hold, then the limit is zero. If this x log(x) moment condition holds, then we prove L 1 convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing. If, in addition, a suitable extra power moment condition on the branching mechanism holds, then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L 1 limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove L 2 convergence of an appropriately scaled process and the above mentioned projections as well. A representation of the limits is also provided under the same moment conditions.

Under a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

No abstract