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

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

2009-03-02

arxiv.org/abs/0903.0255v1

Annals of Applied Probability 2009, Vol. 19, No. 1, 186-209

Mathematics

Probability

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

Scientific paper

10.1214/08-AAP538

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.

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