  We prove that the harmonic measure is stationary, unique and invariant on the
interface of DLA growing on a cylinder surface. We provide a detailed
theoretical analysis puzzling together multiscaling, multifractality and
conformal invariance, supported by extensive numerical simulations of clusters
built using conformal mappings and on lattice. The growth properties of the
active and frozen zones are clearly elucidated. We show that the unique scaling
exponent characterizing the stationary growth is the DLA fractal dimension.
