A general formalism for hopping electron transport over insulators sustained by secondary electron emission is presented. Steady-state electron transport takes place when the charging of the insulator, which turns out to be self-stabilizing, is such that the average secondary electron yield becomes equal to unity. The steady-state potential distribution for the electron transport is determined for various insulating geometries with the aid of Monte Carlo calculations and compared with the low-hopping approximation. The Monte Carlo results show that the steady-state potential distribution can exhibit several interesting features such as spontaneous symmetry breaking of statistical origin and the occurrence of local repulsive parts in the geometry. In several cases the numerical results, including the above two features, are found to agree well with the results of experiments.