Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations

TL;DR

This paper investigates the long-term behavior of solutions to a general one-dimensional inelastic kinetic equation, establishing conditions for convergence to equilibrium and characterizing the equilibrium as a mixture of stable laws.

Contribution

It provides necessary and sufficient conditions for convergence to equilibrium in inelastic kinetic equations, linking initial data to stable law domains of attraction.

Findings

Solutions converge to equilibrium if initial data is in the domain of attraction of a stable law.

Equilibrium solutions can be expressed as mixtures of stable laws.

The analysis uses a generalized CLT applied to Skorokhod representations.

Abstract

For a general inelastic Kac-like equation recently proposed, this paper studies the long-time behaviour of its probability-valued solution. In particular, the paper provides necessary and sufficient conditions for the initial datum in order that the corresponding solution converges to equilibrium. The proofs rest on the general CLT for independent summands applied to a suitable Skorokhod representation of the original solution evaluated at an increasing and divergent sequence of times. It turns out that, roughly speaking, the initial datum must belong to the standard domain of attraction of a stable law, while the equilibrium is presentable as a mixture of stable laws.