Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View

Regazzini, Eugenio

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 175-198

Résumé

Let $f(\cdot, t)$ be the probability density function representing the solution of Kac's Boltzmann-like 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 $f_{0}$. Henry McKean Jr. put forward the conjecture 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 $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture, and concludes with the theorem stating that this holds true whenever $f_{0}$ has finite fourth moment and its Fourier transform $\varphi_{0}$ satisfies $|\varphi_{0}(\xi)| = o(|\xi|^{-p})$ as $|\xi| \to + \infty$, for some $p > 0$. The first part of the lecture expounds the derivation of the Kac Boltzmann-like equation from the Kac master equation. A detailed description of the probabilistic methods resorted to prove the above-mentioned theorem is then given. The final part mentions further applications of these methods to other kinetic models.