  The scaling invariance for chaotic orbits near a transition from unlimited to
limited diffusion in a dissipative standard mapping is explained via the
analytical solution of the diffusion equation. It gives the probability of
observing a particle with a specific action at a given time. We show the
diffusion coefficient varies slowly with the time and is responsible to
suppress the unlimited diffusion. The momenta of the probability are determined
and the behavior of the average squared action is obtained. The limits of small
and large time recover the results known in the literature from the
phenomenological approach and, as a bonus, a scaling for intermediate time is
obtained as dependent on the initial action. The formalism presented is robust
enough and can be applied in a variety of other systems including time
dependent billiards near a transition from limited to unlimited Fermi
acceleration as we show at the end of the letter and in many other systems
under the presence of dissipation as well as near a transition from
integrability to non integrability.
