The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, $I$, and angle, $\theta$ and controlled by two control parameters: (i) $\epsilon$, controlling the nonlinearity of the system, particularly a transition from integrable for $\epsilon=0$ to non-integrable for $\epsilon\ne 0$ and; (ii) $\gamma$ denoting the power of the action in the equation defining the angle. For $\epsilon\ne 0$ the phase space is mixed and
chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures. For the chaotic dynamics far apart from the periodic islands, normal diffusion is observed. The scenario changes significantly when the dynamics passes near stability regions where anomalous diffusion dominates over the dynamics, stickiness is present and a temporary break of ergodicity is observed.