We use probabilistic methods to study properties of mean-field models, which arise as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle ‘jumps forward’ at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles’ distribution when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of—and attraction to—mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Inspired by a PDE–ODE system of aggregation developed in the biomathematical literature, we investigate an interacting particle system representing aggregation at the level of individuals. We prove that the empirical density of the individual converges to the solution of the PDE–ODE system.

This paper considers estimators of parameters of the Boolean model which are obtained by means of the method of intensities. For an estimator of the intensity of the point process of germ points the asymptotic normality is proved and the corresponding variance is given. The theory is based on a study of second-order characteristics of the point process of lower-positive tangent points of the Boolean model. An estimator of the distribution of a typical grain is also discussed.