A Wright-Fisher model with indirect selection

We study a generalization of the Wright--Fisher model in which some individuals adopt a behavior that is harmful to others without any direct advantage for themselves. This model is motivated by studies of spiteful behavior in nature, including several species of parasitoid hymenoptera in which sperm-depleted males continue to mate de- spite not being fertile. We first study a single reproductive season, then use it as a building block for a generalized Wright--Fisher model. In the large population limit, for male-skewed sex ratios, we rigorously derive the convergence of the renormalized process to a diffusion with a frequency-dependent selection and genetic drift. This allows a quantitative comparison of the indirect selective advantage with the direct one classically considered in the Wright--Fisher model. From the mathematical point of view, each season is modeled by a mix between samplings with and without replacement, and analyzed by a sort of "reverse numerical analysis", viewing a key recurrence relation as a discretization scheme for a PDE. The diffusion approximation is then obtained by classical methods

Ergodicity of the zigzag process

The zigzag process is a Piecewise Deterministic Markov Process which can be used in a MCMC framework to sample from a given target distribution. We prove the convergence of this process to its target under very weak assumptions, and establish a central limit theorem for empirical averages under stronger assumptions on the decay of the target measure. We use the classical "Meyn-Tweedie" approach. The main difficulty turns out to be the proof that the process can indeed reach all the points in the space, even if we consider the minimal switching rates

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology

Should I Stay or Should I Go?: Why Bolivian Tactics and U.S. Flexibility Undermine the Single Convention on Narcotic Drugs

This Note examines the deterioration of the 1961 Single Convention on Narcotic Drugs (“Convention”), as Member State circumvention of treaty spirit continues to highlight the disconnect between progressive drug policies and an enforcement regime armed with little or no power to monitor compliance. It first provides a brief history of the Convention with discussion underscoring the governing bodies of the treaty itself, particularly the International Narcotics Control Board (INCB). Further discussion will note the procedural mechanisms whereby parties can propose amendments or reservations to the Convention itself. With such procedures in mind, subsequent examination will analyze how Member States have come to skirt their responsibilities owed to the Convention, particularly Bolivia’s withdrawal and reaccession and the United States’ “flexibility” approach in interpreting the guidelines of the treaty. Additionally, this Note will examine what threats and problems have been posed to the international narcotics community and to the Convention itself as a result of such circumvention and the inability of the INCB to effectively ensure strict compliance of the treaty. In light of these problems, this Note will highlight several possible solutions to the problem of circumvention, including a blanket amendment preventing party withdrawal and reaccession with reservation and empowering governing bodies like the INCB so that compliance may be ensured in more effective fashion

Functional inequalities and uniqueness of the Gibbs measure -- from log-Sobolev to Poincar\'e

In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincar\'e. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at

Gérer le risque dans une métropole : le système français face à l'inondation dans l’agglomération parisienne

Le propos de cet article est de montrer qu’une des principales difficultés de gestion des inondations en ville réside dans la faible prise en compte des interactions entre le risque et la spécificité de l'espace urbain. La situation est encore plus complexe dans le cas de l'agglomération parisienne, où les dynamiques métropolitaines viennent s’ajouter aux dynamiques urbaines classiques. Il s’agit donc d'étudier les outils de gestion existants ainsi que les systèmes d’acteurs auxquels ils renvoient, afin de souligner les limites du dispositif français et de proposer des pistes de réflexion pour l’élaboration d’une gouvernance des inondations qui intègre le risque dans les dynamiques territoriales de la métropole.The focus of this paper is to discuss the difficulties with flood hazard management in urban areas. The main problem is that interactions between flood risk and spatial characteristics of cities are barely taken into consideration. The case of the Parisian metropolitan area is even more complicated as metropolitan and local urban dynamics combine. We examine the management tools currently being used and the system of actors involved to show the limits of the French operations. We then explore ways to develop a governance framework that takes flood hazard into account when considering metropolitan land development

Resonances for a diffusion with small noise

We study resonances for the generator of a diffusion with small noise in $R^d$ :$L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.Comment: 36