REACHING THE BEST POSSIBLE RATE OF CONVERGENCE TO EQUILIBRIUM FOR SOLUTIONS OF KAC'S EQUATION VIA CENTRAL LIMIT THEOREM

• 成果类型: Article
• 署名作者: Dolera, Emanuele; Gabetta, Ester; Regazzini, Eugenio
• 署名单位: University of Pavia
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP538
• 发表日期: 2009
• 页码: 186-209
• 关键词: maxwellian molecules

摘要

Let f (.,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(., t) and g(sigma) goes to zero, as t ->+infinity, 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 phi(0) satisfies vertical bar phi(0)(xi)vertical bar = o(vertical bar xi vertical bar(-P)) as vertical bar xi vertical bar -> +infinity, 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.