A Galton-Watson process in a varying environment is a discrete time branching process where the offspring distributions vary among generations. It is known that in the critical case, these processes present a Yaglom limit, that is, a suitable normalization of the process conditioned on non-extinction converges in distribution to a standard exponential random variable. In this manuscript, we provide the rate of convergence of the Yaglom limit with respect to the Wasserstein metric.

The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of this process on Galton-Watson trees, where vertices are equipped with a random fitness representing inhomogeneities among individuals. In this paper, we establish conditions under which the contact process with fitness on Galton-Watson trees exhibits a phase transition. We prove that if the distribution of the product of the offspring and fitness has exponential tails then the survival threshold is strictly positive. Further, we show that, under certain conditions on either the fitness distribution or the offspring distribution, there is no phase transition and the process survives with positive probability for any choice of the infection parameter. A similar dichotomy is known for the contact process on a Galton-Watson tree. However, we see that the introduction of fitness means that we have to take into account the combined effect of fitness and offspring distribution to decide which scenario occurs.

In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to −∞ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to −∞ or oscillates. We extend recent results of Palau et al. [25] and Li and Xu [22], where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. [4] where the critical case was studied. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes and the asymptotic behaviour of exponential functionals of Lévy processes. As an application of the aforementioned results, we characterise the process conditioned to survival also known as the Q-process.