Central limit theorem for a class of one-dimensional kinetic equations

TL;DR

This paper studies a class of one-dimensional Boltzmann equations, extending classical models, and proves their solutions converge over time to a limit distribution related to stable laws, with explicit rates under certain conditions.

Contribution

It introduces a new class of Boltzmann equations linked to the central limit theorem and establishes convergence to stable law mixtures with explicit exponential rates.

Findings

Solutions converge to a mixture of stable laws.

Explicit exponential convergence rates are derived.

Strong convergence of densities is demonstrated.

Abstract

We introduce a class of Boltzmann equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with quite general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions towards a limit distribution. If the initial condition for the Boltzmann equation belongs to the domain of normal attraction of a certain stable law $g_\alpha$, then the limit is non-trivial and is a statistical mixture of dilations of $g_\alpha$. Under some additional assumptions, explicit exponential rates for the equilibration in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.