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

Sebastian Mentemeier; Glib Verovkin

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arXiv · 2501.05133

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

Sebastian Mentemeier·Glib Verovkin

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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.

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Sebastian Mentemeier, Glib Verovkin. 2025-01-09. A probabilistic study of the set of stationary solutions to spatial kinetic-type equations. https://arxiv.org/abs/2501.05133

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