of the gain in regularity that should result from the Fokker-Planck term, we iterate in a weighted $L^2$ setting, estimating moments $v_0^\gamma \partial_{x,v}^k f$ in $L^2$. Other crucial features which appear include the cone estimate, conservation of mass, and the symmetry and positivity of the diffusive operator.

Though this is the first investigation of the well-posedness of (\[RVMFP\]), others have studied Vlasov-Maxwell models incorporating a Fokker-Planck term. Both Yu and Yang [@YangYu] and Chae [@Chae] constructed global classical solutions to the non-relativistic Vlasov-Maxwell-Fokker-Planck system for initial data sufficiently close to Maxwellian using Kawashima estimates and the well-known energy method. Additionally, Lai [@Lai; @Lai2] arrived at a similar result for a one and one-half dimensional “relativistic” Vlasov-Maxwell-Fokker-Planck system using classical estimates. The unfortunate commonality amongst these models, however, is that they lack invariance properties. Namely, each couples the Lorentz-invariant Maxwell equations to either a Galilean-invariant Vlasov equation with non-relativistic velocities or a hybrid Vlasov equation that includes relativistic velocity corrections, but utilizes the Laplacian $\Delta_v$ as the Fokker-Planck term. This latter term destroys the inherent Lorentz-invariance of the relativistic Vlasov-Maxwell system. Thus, we consider a diffusive operator of the form $\nabla_v \cdot (D \nabla_v f)$ which preserves this property. With this 