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

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 $ \mathbb{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 [18] and of its interpretation by McKean [10,11].