Identification and existence of Boltzmann processes

Abstract

The stochastic differential equation of McKean-Vlasov type is identified such that the Fokker-Planck equation associated to it is the Boltzmann equation. Hence, we call its solutions as Boltzmann processes. They describe the dynamics (in position and velocity) of particles expanding in vacuum in accordance with the Boltzmann equation. Given a solution $f:=$ $\{f(t,x,v{\}}_{0\le t\le T}$ of the Boltzmann equation, the existence of solutions to the McKean-Vlasov SDE is established for the non-cutoff hard sphere case.