Central limit theorem for the solution of the Kac equation

Details Central limit theorem for the solution of the Kac equation Central limit theorem for the solution of the Kac equation

2009-01-16

arxiv.org/abs/0901.2464v1

Annals of Applied Probability 2008, Vol. 18, No. 6, 2320-2336

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP524 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP524

We prove that the solution of the Kac analogue of Boltzmann's equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof of the convergence to the Maxwellian distribution is provided, with a rate information both under the sole hypothesis that the initial energy is finite and under the additional condition that the initial distribution has finite moment of order $2+\delta$ for some $\delta$ in $(0,1]$. Moreover, it is proved that finiteness of initial energy is necessary in order that the solution of Kac's equation can converge weakly. While this statement may seem to be intuitively clear, to our knowledge there is no proof of it as yet.

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