A drift‐kinetic Eulerian Vlasov code, with fluid equations for the ions, is developed to study the problem of the injection of an electron beam into a two‐dimensional magnetized plasma, often referred to as direct current (dc) helicity injection. The diffusion of electrons across a magnetic field in the presence of a beam–plasma instability is studied. The case of a magnetic field tilted with respect to the beam direction is considered. The competition between the velocity shear Kelvin–Helmholtz (KH) and the beam–plasma (BP) instabilities is investigated in order to analyze the plasma heating and current drive mechanism induced by the beam injection. The KH instability generates low‐frequency plasma convection motion associated withcE×B/B2drift. In particular, the diffusion coefficientsDyandDv∥describing, respectively, the anomalous diffusion process induced in space across the magnetic field by the KH instability, and the velocity diffusion process due to the kinetic effects induced in velocity space along the magnetic field by the BP instability, are computed using test‐particle diagnostics. In the present Cartesian model, it is found thatDy=Dv∥ tan2 &thgr;/&ohgr;ce2where &thgr; is the angle between the magnetic field and thexaxis. This relation which links the electron dynamics in thex‐yreal space and in thex‐v∥phase space is verified by the numerical code. The Vlasov code provides a powerful tool to study particle diffusion in space and in phase space, especially in the low‐density regions of the distribution function.