Propagation of chaos for interacting particles subject to environmental noise

A system of interacting particles described by stochastic differential equations is considered. As oppopsed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (noninteracting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of inviscid type, as opposed to the case when independent noises drive the different particles.Comment: Published at http://dx.doi.org/10.1214/15-AAP1120 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

An evolutionary model for simple ecosystems

In this review some simple models of asexual populations evolving on smooth landscapes are studied. The basic model is based on a cellular automaton, which is analyzed here in the spatial mean-field limit. Firstly, the evolution on a fixed fitness landscape is considered. The correspondence between the time evolution of the population and equilibrium properties of a statistical mechanics system is investigated, finding the limits for which this mapping holds. The mutational meltdown, Eigen's error threshold and Muller's ratchet phenomena are studied in the framework of a simplified model. Finally, the shape of a quasi-species and the condition of coexistence of multiple species in a static fitness landscape are analyzed. In the second part, these results are applied to the study of the coexistence of quasi-species in the presence of competition, obtaining the conditions for a robust speciation effect in asexual populations.Comment: 36 pages, including 16 figures, to appear in Annual Review of Computational Physics, D. Stauffer (ed.), World Scientific, Singapor

Mean field limit of interacting filaments and vector valued non linear PDEs

Families of $N$ interacting curves are considered, with long range, mean field type, interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as $N$ goes to infinity, to a mean field current, solution of a nonlinear, vector valued, partial differential equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero. This set-up is inspired from vortex filaments in turbulent fluids, although for technical reasons we have to restrict to smooth interaction, instead of the singular Biot-Savart kernel. All these results are based on a careful analysis of a nonlinear flow equation for 1-currents, its relation with the vector valued PDE and the continuous dependence on the initial conditions

Social Security and Retirement in Spain

We describe the historical evolution of the Spanish Social Security system and its current organization. Our attention concentrates on the main public pension scheme for private employees in the manufacturing and service sector (RGSS) which covers by far the largest majority of Spanish workers. After describing the way in which pension and retirement decisions are regulated by this system, we try to compute the incentives to early retirement it provides to different kinds of individuals. We show that the Spanish SS legislation generates strong incentives to retire early and that Spanish workers tend to do so. In particular, our findings support the idea that pensions-induced incentives matter for the labor supply behavior of" Spanish workers. While the Spanish system does not pay a particularly generous average pension relative to GDP per-capita, its generosity' concentrates on providing large minimum pensions to individuals with below average working histories and/or low wages. At the same time, the pension system provides workers earning average or above average salaries and with complete working histories with relatively weak financial gains from not retiring after the age of 60. The combination of these features of the Spanish legislation seems to account well for the observed increase in the percentage of early retirees among Spanish pensioners during the nineties.

Social security and retirement in Spain.

This paper examines the interplay between the retirement incentives generated by the Spanish public pension system and the decision to early retire from the labor force. For many workers, particularly those with below--median earning profiles or with incomplete working histories, retirement at ages earlier then 65 appears to be the most rational strategy. Indeed we calculate that, exception made for workers in the upper twenty percent of the earnings distribution, the current system generates strong incentives to retire before the age of 65. We identify the minimum pension rule, embedded in the Spanish legislation, as the most important source of such incentives. We apply the insights from our study to a preliminary evaluation of the reform legislation just enacted (July 26, 1997). We find it doubtful that it may seriously weaken such incentives. On the contrary, the final outcome may be an increase in the proportion of workers for which the minimum pension is binding, thereby increasing the incentive to early retirement.Retirement; Pensions; incentives;