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

## TL;DR

This paper introduces a recursive algorithm and a series expansion for the homogeneous Boltzmann equation with hard potentials, providing explicit representations of solutions based on initial data, though practical efficiency is limited.

## Contribution

It presents a novel recursive algorithm and a series expansion for solutions to the Boltzmann equation with hard potentials, extending Wild's sum to non-Maxwellian molecules.

## Key findings

- The recursive algorithm produces a random variable representing the solution.
- The series expansion explicitly expresses the solution in terms of initial data.
- Both methods are theoretically interesting but may be impractical for computations.

## Abstract

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\infty)\times\mathbb{R}^3$ random variable $(M_t,V_t)$ such that $E[M_t {\bf 1}_{\{V_t \in \cdot\}}]=f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express $f_t$ in terms of $f_0$, 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 and of its interpretation by McKean.