The problem of the electron immersed in the random zeropoint radiation field and described by the stochastic Abraham-Lorentz equation is analysed from a new point of view. First an approximate treatment of the (statistically) stationary motion of this system is performed by using a local linearization procedure applicable to nonlinear periodic problems. Our treatment leads to a (quantum) condition on the mean kinetic energy of the stationary states. The random field is thus seen to have a selective and stabilizing effect on the dynamics. The results are applied to the hydrogen atom and other problems and the old quantum rules are briefly discussed in the light of this quantization mechanism. In the second part we develop a formalism to describe the stationary regime of this stochastic system. Using the canonical properties of the vacuum field amplitudes, we derive symplectic relations for the particle variablesx, p; in addition, a canonical treatment of the system allows us to derive the equations of evolution. A Hilbert-space formalism arises as a possible tool for the mathematical treatment (inx orp space) and it is shown to lead to the usual description of quantum mechanics.