Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

Markov jump processes and collision-like models in the kinetic description of multi-agent systems

Pages: 1539 – 1568



Nadia Loy (Department of Mathematical Sciences, Politecnico di Torino, Italy)

Andrea Tosin (Department of Mathematical Sciences, Politecnico di Torino, Italy)


Multi-agent systems can be successfully described by kinetic models, which allow one to explore the large scale aggregate trends resulting from elementary microscopic interactions. The latter may be formalised as collision-like rules, in the spirit of the classical kinetic approach in gas dynamics, but also as Markov jump processes, which assume that every agent is stimulated by the other agents to change state according to a certain transition probability distribution. In this paper we establish a parallelism between these two descriptions, whereby we show how the understanding of the kinetic jump process models may be improved taking advantage of techniques typical of the collisional approach.


transition probability, Boltzmann-type equation, quasi-invariant limit, Fokker–Planck equation, Maxwellian, Monte Carlo algorithm

2010 Mathematics Subject Classification

35Q20, 35Q70, 35Q84

Received 23 May 2019

Accepted 17 March 2020

Published 4 November 2020