Kinetic models for systems of interacting agents with multiple microscopic states

TL;DR

This paper introduces general kinetic models for systems of interacting agents with multiple microscopic states, applicable to socio-economic phenomena, and analyzes their mathematical properties, asymptotic behavior, and numerical solutions.

Contribution

It develops a broad class of kinetic models with transition probabilities for multiple state changes, including mathematical analysis and numerical simulations.

Findings

Existence and uniqueness of solutions in Wasserstein spaces.

Asymptotic regimes leading to Fokker-Planck equations.

Numerical demonstrations of distribution evolution.

Abstract

We propose and investigate general kinetic models %of Boltzmann type with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. These models can be applied to many problems in socio-economic sciences, where individuals may change both their compartment and their characteristic kinetic variable, as for instance kinetic models for epidemics or for international trade with possible transfers of agents. Mathematical properties of our kinetic model are proved, as existence and uniqueness of a solution for the Cauchy problem in suitable Wasserstein spaces. The quasi-invariant asymptotic regime, leading to simpler kinetic Fokker-Planck-type equations, is investigated and commented on in comparison with other existing models. Some numerical tests are performed in order to show time evolution of distribution functions and of meaningful macroscopic fields, even in case of non-constant interaction probabilities.