We present new solutions of the pitch-angle-averaged transport equation of pick-up ions in the heliosphere. These solutions supplement those available so far for the case of a vanishing momentum diffusion. A general solution is derived for nonvanishing momentum diffusion. This solution, which is valid for a simple spatial power law behavior of the solar wind plasma and for arbitrary source functions, is applied to the stochastic acceleration of pick-up ions and generalizes earlier solutions. Based on this, we present several asymptotic phase space integrals for the outer heliosphere and solve them analytically for a standard pick-up ion source function. Numerical calculations for interstellar neutral hydrogen yield phase space distributions depending sensitively on the underlying turbulence and its dependence on heliocentric distance.