A weak criterion of absolute continuity for jump processes: application to the Boltzmann equation

Abstract

We first prove a general and quite simple criterion of absolute continuity, based on the use of almost sure derivatives, which is applicable even when integration by parts may not be used. We apply it to Poisson-driven stochastic differential equations. Next, using a typically probabilistic substitution in the Boltzmann equation, we extend Tanaka's probabilistic interpretation for spatially homogeneous Boltzmann equations with Maxwell molecules and without angular cut-off to much more general spatially homogeneous two-dimensional Boltzmann equations. We relate a measure-solution {Q_t}_t of the equation to a solution V_t of a nonlinear Poisson-driven stochastic differential equation: for each t, Q_t is the law of V_t. We extend our absolute continuity criterion to these nonlinear Poisson functionals and prove that even in the case of degenerate initial distribution, the law of V_t admits a density f(t,⋅) for each t>0, which is hence a solution to the Boltzmann equation. We thus obtain an original existence result.