Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of Boltzmann equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with quite general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions towards a limit distribution. If the initial condition for the Boltzmann equation belongs to the domain of normal attraction of a certain stable law nu_alpha, then the limit is a scale mixture of nu_alpha. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown. Keywords:Central limit theorem, Boltzmann equation, Domains of normal attraction, Kac model, Smoothing transformations, Stable law, Sums of weighted independent random variables. AMS Subject Classification: 60F05, 82C40.