A probabilistic study of the set of stationary solutions to spatial kinetic-type equations
Probability
2025-01-10 v1
Abstract
In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Using a representation of the collision operator derived in Bassetti, Ladelli, Matthes (2015) and Dolera, Regazzini (2014), we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under assumptions as weak as possible. Our main objective is a characterisation of the set of stationary solutions, e.g. equilibrium solutions for inelastic kinetic-type equations, which we describe as mixtures of multidimensional stable laws.
Keywords
Cite
@article{arxiv.2501.05133,
title = {A probabilistic study of the set of stationary solutions to spatial kinetic-type equations},
author = {Sebastian Mentemeier and Glib Verovkin},
journal= {arXiv preprint arXiv:2501.05133},
year = {2025}
}