Rescaling stochastic processes: Asymptotics

In this chapter the authors investigate the links among different scales, from a probabilistic point of view. Particular attention is being paid to the mathematical modelling of the social behavior of interacting individuals in a biological population, on one hand because there is an intrinsic interest in dynamics of population herding, on the other hand since agent based models are being used in complex optimization problems. Among other interesting features, these systems lead to phenomena of self-organization, which exhibit interesting spatial patterns. Here we show how properties on the macroscopic level depend on interactions at the microscopic level; in particular suitable laws of large numbers are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction–diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large. As a working example, an interacting particle system modelling social behavior has been proposed, based on a system of stochastic differential equations, driven by both aggregating/repelling and external “forces”. In order to support a rigorous derivation of the asymptotic nonlinear integro-differential equation, compactness criteria for convergence in metric spaces of measures, and problems of existence of a weak/entropic solution have been analyzed. Further the temporal asymptotic behavior of the stochastic system of a fixed number of interacting particles has been discussed. This leads to the problem of the existence of nontrivial invariant probability measure.