On global existence and trend to the equilibrium for the Vlasov–Poisson–Fokker–Planck system with exterior confining potential

We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov–Poisson–Fokker–Planck system with small non-linear term but with a possibly large exterior confining potential in dimension d=2d=2 and d=3d=3. The proof relies on a fixed point argument using sharp estimates (at short and long time scales) of the semi-group associated to the Fokker–Planck operator, which were obtained by the first author.