The coronal magnetic field is subject to random footpoint motions that cause small-scale twisting and braiding of field lines. We present a mean field theory describing the effects of such small-scale twists on the large-scale coronal field. This theory assumes that the coronal field is force free, with electric currents flowing parallel or antiparallel to magnetic field lines. Random footpoint motions are described in terms of diffusion of the mean magnetic field at the photosphere. The appropriate mean field equations are derived, and a numerical method for solving these equations in three dimensions is presented. Preliminary results obtained with this method are also presented. In particular the formation of filament channels is studied. Filament channels are regions where the coronal magnetic field is strongly aligned with the underlying polarity inversion line in the photosphere. It is found that magnetic flux cancellation plays an important role in the formation of such channels. Various models of the coronal field are presented, including some in which the axial field is assumed to originate from below the photosphere. The models reproduce many of the observed features of filament channels, but the observed hemisphere pattern of dextral and sinistral channels remains a mystery.