Boltzmann-like kinetic models and Boltzmann maps

This work shows how Boltzmann maps can be used in the study of a class of kinetic models formally similar to those recently proposed in population dynamics. Boltzmann maps are nonlinear, discrete, doubly-stochastic processes which have been introduced in statistical physics for the description of nonequilibrium phenomena. The key feature of Boltzmann maps is that they increase the entropy, when applied to nonstationary states, if their interaction operators have a spectral gap. This fact can be used to analyse the asymptotic behaviour of the relevant dynamics. In particular, in some circumstances we can prove that stationary states exist, are unique, and stable. Moreover, we can supplement these results with theorems on the global convergence of the dynamics to such stationary states.