The statistical behavior of the domain wall depinning from a notch placed in a thin ferromagnetic wire is studied by means of a stochastic one-dimensional model which considers the wall as a rigid object inside a parabolic potential at room temperature. This analysis reveals the key role of thermal fluctuations on the current and field-induced domain wall depinning, and it allows for direct comparison with experiments in order to gain information on the nonadiabaticity degree of the spin torque.