[en] Transport in a stochastic magnetic layer is an important issue in tokamak physics which finds a stimulating counterpart in experiments where the stochasticity is controlled by a set of specific coils, as in Tore Supra (Ergodic Divertor) or on TEXT (Ergodic Magnetic Limiter). In this paper the authors first recall the results obtained for particle transport. Using the same method of functional extremum calculation they investigate the regime of weak stochasticity due to the finite connection length to the wall L_wall. In this case one has L_wall approx-lt L_K where L_K is the Kolmogorov scale of the stochastic magnetic field. This regime is readily expected in the experiments and governs the heat deposition through the so-called laminar domain. In the ergodic domain they use again this method to calculate the effective resistivity along the toroidal direction. They find that in all cases there is a strong increase of this resistivity which will account for the narrowing of the current channel and eventually for the stabilization of the MHD (2.1) mode