Global Cauchy problem for the Vlasov–Riesz–Fokker–Planck system near the global Maxwellian

We prove the global existence and uniqueness of solutions to the Vlasov–Riesz–Fokker–Planck system around the global Maxwellian in the periodic spatial domain. Depending on the order of Riesz potential, we present two frameworks for the construction of global-in-time solutions with Sobolev and analytic regularity. The analytic function framework covers the Vlasov–Dirac–Benney–Fokker–Planck system. Furthermore, we show the exponential decay of solutions toward the global Maxwellian. Our result is generalized to the whole space case in which the decay rate of convergence is algebraic.