Planetary migration can happen when planets are young as they scatter planetesimal debris; it can also follow a period of dynamical upheaval. The outward migration of Neptune from an original location closer to the Sun provides a likely explanation for the prevalence of Kuiper belt objects (KBOs) currently in mean motion resonance with the planet. However, resonance capture and retention by a migrating Neptune require the planet’s migration to be sufficiently smooth. Planetesimal-driven migration is stochastic (noisy), particularly when the planetesimals driving migration are large. We confirm a previously derived analytic model of stochasticity in planetary migration through numerical simulations, and we determine the values of unknown coefficients in this model. We then identify the relative impact of noisy migration on resonance retention for all resonances up to fourth order lying between the 3:2 and the 3:1. The current mass of the Kuiper belt is 100-1000 times less massive with respect to what models suggest was present in the region of Neptune’s formation and outward migration. It is still uncertain what physical processes caused such a substantial depletion of material. We use the current observed size distribution of the dynamically hot population with a break radius of ~70 km to find a maximum constraint on the amount of mass in large (r ~ 500 km) objects during any epoch of Neptune’s migration. We calculate the characteristic loss time of several resonances and find that the population of large objects could not have been more than 1000 times more massive than it is now, by the end of Neptune’s evolution.