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

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.