A discontinuity solution of the impurity diffusion equation is derived for a local pulsed source to yield the finite impurity propagation velocity at the initial stage. At the final stage the diffusion equation is shown to reduce to the Einstein equation. A theoretical model for the space-time distribution of the avalanche-multiplied electrons is proposed. The model describes major mechanisms responsible for the evolution of the optical discharge in aerosol medium from its initiation to the shock-wave formation.