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

Abstract

Let f(⋅, t) be the probability density function which represents the solution of Kac’s equation at time t, with initial data f0, and let gσ be the Gaussian density with zero mean and variance σ2, σ2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f(⋅, t) and gσ goes to zero, as t→+∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform ϕ0 satisfies |ϕ0(ξ)|=o(|ξ|−p) as |ξ|→+∞, 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.