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 g_σ 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 g_σ 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.