The mean velocity of a Brownian sphere sedimenting under the influence of gravity through a viscous fluid confined within a vertical circular cylinder whose radius changes periodically along its effectively infinite length is investigated for circumstances where, owing to the small radius of the sphere relative to the minimum cylinder radius, conventional hydrodynamic wall effects are absent. Using generalized Taylor dispersion theory, explicit results are presented for the case where the periodic changes in cylinder radius occur abruptly, representing the computationally and conceptually simplest configuration. Despite the apparent absence of wall effects, the (mean) settling velocity of the sphere is always less than its Stokes law value, and depends nonlinearly upon the gravity force. Moreover, the sphere's dispersivity fails to obey a Stokes-Einstein relation. This anomalous behavior may be exploited to enhance the force-driven separation of physicochemically disparate Brownian particles.