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

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

2011-03-24

arxiv.org/abs/1103.4738v1

Mathematics

Probability

Scientific paper

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

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