A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff

Abstract

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution \begin{document}$ (f_t)_{t\geq 0} $\end{document} , once the initial condition \begin{document}$ f_0 $\end{document} with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a \begin{document}$ (0,\infty)\times {\mathbb{R}}^3 $\end{document} random variable \begin{document}$ (M_t,V_t) $\end{document} such that \begin{document}$ \mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t $\end{document} . We also write down a series expansion of \begin{document}$ f_t $\end{document} . Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express \begin{document}$ f_t $\end{document} in terms of \begin{document}$ f_0 $\end{document} , we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [ 18 ] and of its interpretation by McKean [ 10 , 11 ].