On the construction and identifcation of Boltzmann processes

Given the existence of a solution f(t; x; v)_{t \in \mathbb{R}_+^0} of the Boltzmann equation for hard spheres, we introduce a stochastic differential equation driven by a Poisson random measure that depends on f(t; x; v). The marginal distributions of its solution solves a linearized Boltzmann equation in the weak form. Further, if the distributions admit a probability density, we establish, under suitable conditions, that the density at each t coincides with f(t; x; v). The stochastic process is therefore called the Boltzmann process.