We study a one-dimensional diffusion X in a drifted Brownian potential W κ , with 0 < κ < 1, and focus on the behavior of the local times (L(t, x), x) of X before time t > 0. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite site. These limits can be written explicitly from a two dimensional stable Lévy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of X.

We study the properties of the exponential functional +∞ 0 e −X ↑ (t) dt where X ↑ is a spectrally one-sided Lévy process conditioned to stay positive. In particular, we study finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at 0 and smoothness of the density.

Consider a bi-allelic population subject to neutral reproduction, genic selection and mutation, which is susceptible to exceptional changes in the environment. Neutral reproductions are modeled as in the classical Wright-Fisher diffusion model, mutation is parent independent and genic selection is reflected by an additional rate at which fit individuals reproduce. Moreover, changes in the environment accentuate the selective advantage of fit individuals. The evolution of the type composition is then described by a Wright-Fisher-type SDE with a jump term, modeling the effect of the environment and involving the stochastic derivative of a subordinator. Our interest in this paper is twofold: on the one side we aim to understand the influence of the environment in the type composition in the population, and on the other hand we aim to reveal the ancestral picture behind this model. The latter is described by means of an extension of the ancestral selection graph (ASG). The relation between forward and backward objects is given via duality. More precisely, we establish annealed and quenched moment dualities between the type-frequency process and the block-counting process of a variant of the ASG. As an application, we obtain a characterization of the moments of the asymptotic type distribution in the annealed and quenched form. In a similar way, we also obtain annealed and quenched results for the ancestral type distribution. In the absence of mutations, one of the types fixates and our results yield expressions for the fixation probabilities.

We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Lévy environment. In particular, we link this behavior with the left tail of an exponential functional of the environment conditioned to stay positive.