$$\text {L}^2$$-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System

This paper is devoted to the linearized Vlasov–Poisson–Fokker–Planck system in presence of an external potential confinement. We investigate large time behaviour solutions using hypocoercivity methods and a notion scalar product adapted Poisson coupling. Our framework provides estimates which are uniform diffusion limit. As application simple case, we study one-dimensional case prove exponential convergence nonlinear without any small mass assumption.