  Stochastic acceleration of charged particles due to interactions with
magnetohydrodynamic (MHD) plasma waves is the dominant process leading to the
formation of the high-energy electron and ion distributions in a variety of
astrophysical systems. Collisions with the waves influence both the
energization and the spatial transport of the particles, and therefore it is
important to treat these two aspects of the problem in a self-consistent
manner. We solve the representative Fokker-Planck equation to obtain a new,
closed-form solution for the time-dependent Green's function describing the
acceleration and escape of relativistic ions interacting with Alfven or
fast-mode waves characterized by momentum diffusion coefficient $D(p)\propto
p^q$ and mean particle escape timescale $t_esc(p) \propto p^{q-2}$, where $p$
is the particle momentum and $q$ is the power-law index of the MHD wave
spectrum. In particular, we obtain solutions for the momentum distribution of
the ions in the plasma and also for the momentum distribution of the escaping
particles, which may form an energetic outflow. The general features of the
solutions are illustrated via examples based on either a Kolmogorov or
Kraichnan wave spectrum. The new expressions complement the results obtained by
Park and Petrosian, who presented exact solutions for the hard-sphere
scattering case ($q=2$) in addition to other scenarios in which the escape
timescale has a power-law dependence on the momentum. Our results have direct
relevance for models of high-energy radiation and cosmic-ray production in
astrophysical environments such as $\gamma$-ray bursts, active galaxies, and
magnetized coronae around black holes.
