Density-valued solutions for the Boltzmann-Enskog process

Abstract The time evolution of a moderately dense gas evolving in vacuum described by the Boltzmann-Enskog equation is studied. The associated stochastic process, the Boltzmann-Enskog process, was constructed in [1] and further studied in [12, 13]. The process is given by the solution of a McKean-Vlasov equation driven by a Poisson random measure with the compensator depending on the distribution of the solution [1, 12]. The existence of a marginal probability density function at each time for the measure-valued solution is established in this article by using a functional-analytic criterion on Besov spaces [8, 11]. In addition to existence, the density is shown to reside in a Besov space. The support of the velocity marginal distribution is shown to be the whole of $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> .