Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem

TL;DR

This paper proves that solutions of Kac's equation converge exponentially fast to a Gaussian equilibrium in total variation distance, under weaker assumptions than previous studies, using a central limit theorem approach.

Contribution

It establishes the exponential convergence rate to equilibrium for Kac's equation with minimal assumptions on initial data, improving upon prior results.

Findings

Total variation distance tends to zero exponentially at rate -1/4.

Convergence proof relies on a central limit theorem approach.

Weaker conditions on initial data than previous literature.

Abstract

Let $f(\cdot,t)$ be the probability density function which represents the solution of Kac's equation at time $t$, with initial data $f_0$, and let $g_{\sigma}$ be the Gaussian density with zero mean and variance $\sigma^2$, $\sigma^2$ being the value of the second moment of $f_0$. This is the first study which proves that the total variation distance between $f(\cdot,t)$ and $g_{\sigma}$ goes to zero, as $t\to +\infty$, with an exponential rate equal to -1/4. In the present paper, this fact is proved on the sole assumption that $f_0$ has finite fourth moment and its Fourier transform $\varphi_0$ satisfies $|\varphi_0(\xi)|=o(|\xi|^{-p})$ as $|\xi|\to+\infty$, for some $p>0$. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.