  The secular evolution of disk galaxies is largely driven by resonances
between the orbits of 'particles' (stars or dark matter) and the rotation of
non-axisymmetric features (spiral arms or a bar). Such resonances may also
explain kinematic and photometric features observed in the Milky Way and
external galaxies. In simplified cases, these resonant interactions are well
understood: for instance, the dynamics of a test particle trapped near a
resonance of a steadily rotating bar is easily analyzed using the angle-action
tools pioneered by Binney, Monari and others. However, such treatments do not
address the stochasticity and messiness inherent to real galaxies - effects
which have, with few exceptions, been previously explored only with complex
N-body simulations. In this paper, we propose a simple kinetic equation
describing the distribution function of particles near an orbital resonance
with a rigidly rotating bar, allowing for diffusion of the particles' slow
actions. We solve this equation for various values of the dimensionless
diffusion strength $\Delta$, and then apply our theory to the calculation of
bar-halo dynamical friction. For $\Delta = 0$ we recover the classic result of
Tremaine & Weinberg that friction ultimately vanishes, owing to the
phase-mixing of resonant orbits. However, for $\Delta > 0$ we find that
diffusion suppresses phase-mixing, leading to a finite torque. Our results
suggest that stochasticity - be it physical or numerical - tends to increase
bar-halo friction, and that bars in cosmological simulations might experience
significant artificial slowdown, even if the numerical two-body relaxation time
is much longer than a Hubble time.
