  We study the consequences of including the running of the QCD coupling in the
equation describing the evolution of the jet quenching parameter $\hat q$ in
the double logarithmic approximation. To start with, we revisit the case of a
fixed coupling, for which we obtain exact solutions valid for generic values of
the transverse momentum (above the medium saturation scale) and corresponding
to various initial conditions. In the case of a running coupling, we construct
approximate solutions in the form of truncated series obtained via successive
iterations, whose convergence is well under control. We thus deduce the
dominant asymptotic behavior of the renormalized $\hat q$ in the limit of a
large evolution time $Y\equiv\ln(L/\lambda)$, with $L$ the size of the medium
and $\lambda$ the typical wavelength of a medium constituent. We show that the
asymptotic expansion is universal with respect to the choice of the initial
condition at $Y=0$ and, moreover, it is remarkably similar to the corresponding
expansion for the saturation momentum of a shockwave (a large nucleus). As
expected, the running of the coupling significantly slows down the increase of
$\hat q$ with $Y$ in the asymptotic regime at $Y\gg 1$. For the
phenomenologically interesting value $Y\simeq 3$, we find an enhancement factor
close to 3, independently of the initial condition and for both fixed and
running coupling.
