A probabilistic study of the set of stationary solutions to spatial kinetic-type equations

TL;DR

This paper investigates the set of stationary solutions to multivariate kinetic equations, including the Boltzmann equation, characterizing equilibrium states as mixtures of multidimensional stable laws using probabilistic methods.

Contribution

It provides a novel probabilistic framework for characterizing stationary solutions of kinetic equations, including inelastic cases, as mixtures of stable laws.

Findings

Existence and uniqueness of time-dependent solutions established.

Stationary solutions characterized as mixtures of multidimensional stable laws.

Applicable to both elastic and inelastic collision models.

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.