Interacting particles systems with delay and random delay differential equations

TL;DR

This paper investigates a kinetic model of active particles with delayed and random delay differential equations, analyzing mean-field limits and different dynamics based on individual trajectories or aggregate data.

Contribution

It introduces a new kinetic model with delay, derives its mean-field limit, and compares dynamics based on full trajectories versus aggregate data.

Findings

Mean-field equations are well-posed via fixed-point methods.

Different dynamics lead to distinct mean-field descriptions.

Potential applications in modeling delayed interactions in particle systems.

Abstract

In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in $\R^d$ in the second case. We analyse in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modelling applications.