  Starting from the dipole representation of small-$x$ evolution we implement
the running of the coupling in a self-consistent way. This results in an
evolution equation for the dipole density in Borel $(b)$ space. We show that
the Borel image of the dipole density is analytic in the neighbourhood of $b=0$
and that it is equal to the BFKL solution at $b=0$. We study the Borel
singularity structure of the dipole cascade emanating from a virtual photon at
small $x$ and find a branch cut on the positive $b$-semiaxis starting at $b=1/
\beta_0$. This indicates the presence of $1/Q^2$ power corrections to the
small-$x$ structure functions. Finally we present numerical results in the
context of D.I.S.
