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Consider the homogeneous Boltzmann equation for Maxwellian molecules. Weprovide a new representation for its solution in the form of expectation of arandom probability measure M. We also prove that the Fourier transform of M isa conditional characteristic function of a sum of independent random variables,given a suitable sigma-algebra. These facts are then used to prove a CLT forMaxwellian molecules, that is the statement of a necessary and sufficientcondition for the weak convergence of the solution of the equation. Such acondition reduces to the finiteness of the second moment of the initialdistribution \mu_0. As a further application, we give a refinement of someinequalities, due to Elmroth, concerning the evolution of the moments of thesolution.

Probabilistic representation for the solution of the homogeneous Boltzmann equation for Maxwellian molecules