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

TL;DR

This paper introduces a probabilistic representation for solutions to the homogeneous Boltzmann equation for Maxwellian molecules, linking it to random measures and establishing a CLT with conditions based on initial moments.

Contribution

It provides a novel probabilistic framework for the Boltzmann equation solution and derives a necessary and sufficient condition for weak convergence based on initial distribution moments.

Findings

A new probabilistic representation of the solution as an expectation of a random measure.

A CLT for Maxwellian molecules with convergence criteria based on second moments.

Refined inequalities for the evolution of moments of the solution.

Abstract

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.