  Iterated conformal mappings are used to obtain exact multifractal spectra of
the harmonic measure for arbitrary Laplacian random walks in two dimensions.
Separate spectra are found to describe scaling of the growth measure in time,
of the measure near the growth tip, and of the measure away from the growth
tip. The spectra away from the tip coincide with those of conformally invariant
equilibrium systems with arbitrary central charge $c\leq 1$, with $c$ related
to the particular walk chosen, while the scaling in time and near the tip
cannot be obtained from the equilibrium properties.
