  We study the electric current in the non-equilibrium Kondo model at zero
magnetic field, using real-time perturbation theory in the Schwinger-Keldysh
formulation. We show that the perturbative coefficients to all orders have a
finite limit at large switch-on time (t_0 to minus infinity), and we give a
prescription for general operators to give finite coefficients in this limit.
We explain how this is related to the fact that the leads play the role of
thermal baths and allow relaxation to occur and the steady state to form. This
proves perturbatively that a steady state is reached in the Schwinger-Keldysh
formulation, and specifies which operators correspond to quantities that have a
well-defined value in the steady state. Then, we show that the steady state can
be described by a special type of density matrix (related to Hershfield's
conjecture for the particular example of the non-equilibrium Kondo model.) In
the second part of the paper we perform a renormalization-group analysis of the
perturbative series. We give a general argument that strongly suggests that the
perturbative series of any average in the steady state satisfies the
equilibrium Callan-Symanzik equations, and show in detail how it works to
one-loop order for the electric current operator inside any average. We finally
compute to two loops order the average of the electric current in the steady
state, and perform a renormalization-group improvement. From this, we give a
universal prescription, valid in the perturbative regime, for comparing the
effect of the electric current to that of the temperature on the ``Kondo
cloud''.
