  We prove the global-in-time existence of large-data finite-energy weak
solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three
space dimensions. The model couples three essential ingredients of magnetized
plasmas: a transport equation for the probability density function, which
models energetic rarefied particles of one species; the incompressible
Navier--Stokes system for the bulk fluid; and a parabolic evolution equation,
involving magnetic diffusivity, for the magnetic field. The physical derivation
of our model is given. It is also shown that the weak solution, whose existence
is established, has nonincreasing total energy, and that it satisfies a number
of physically relevant properties, including conservation of the total
momentum, conservation of the total mass, and nonnegativity of the probability
density function for the energetic particles. The proof is based on a one-level
approximation scheme, which is carefully devised to avoid increase of the total
energy for the sequence of approximating solutions, in conjunction with a weak
compactness argument for the sequence of approximating solutions. The key
technical challenges in the analysis of the mathematical model are the
nondissipative nature of the Vlasov-type particle equation and passage to the
weak limits in the multilinear coupling terms.
