  A residual planetesimal disk of mass 10-100 Earth masses remained in the
outer solar system following the birth of the giant planets, as implied by the
existence of the Oort cloud, coagulation requirements for Pluto, and
inefficiencies in planet formation. Upon gravitationally scattering
planetesimal debris, planets migrate. Orbital migration can lead to resonance
capture, as evidenced here in the Kuiper and asteroid belts, and abroad in
extra-solar systems. Finite sizes of planetesimals render migration stochastic
("noisy"). At fixed disk mass, larger (fewer) planetesimals generate more
noise. Extreme noise defeats resonance capture. We employ order-of-magnitude
physics to construct an analytic theory for how a planet's orbital semi-major
axis fluctuates in response to random planetesimal scatterings. To retain a
body in resonance, the planet's semi-major axis must not random walk a distance
greater than the resonant libration width. We translate this criterion into an
analytic formula for the retention efficiency of the resonance as a function of
system parameters, including planetesimal size. We verify our results with
tailored numerical simulations. Application of our theory reveals that capture
of Resonant Kuiper belt objects by a migrating Neptune remains effective if the
bulk of the primordial disk was locked in bodies having sizes < O(100) km and
if the fraction of disk mass in objects with sizes > 1000 km was less than a
few percent. Coagulation simulations produce a size distribution of primordial
planetesimals that easily satisfies these constraints. We conclude that
stochasticity did not interfere with, nor modify in any substantive way,
Neptune's ability to capture and retain Resonant Kuiper belt objects during its
migration.
