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 ${\phi }_0$ satisfies $|{\phi }_0(\xi )|=o(|\xi {|}^{-p})$ 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.

Published in at http://dx.doi.org/10.1214/08-AAP538 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Central limit theorem for the solution of the Kac equation

Cited by in corpus (2)

• Probabilistic representation for the solution of the homogeneous Boltzmann equation for Maxwellian molecules
• Central limit theorem for a class of one-dimensional kinetic equations