Effects from turbulence-induced fluctuations in water vapor saturation on cloud droplet growth are examined using a Brownian diffusion model. The model predicts diffusive broadening of the droplet size distribution, tempered by enhanced evaporation-induced drift of droplets to smaller size from vapor depletion, and approach to a stationary condition (Weibull distribution) determined by the balance between size-space diffusion and drift. Monte Carlo simulations of the approach to the stationary limit and of the distribution itself are presented and compared favorably with observation. A key turbulence parameter required by the kinetic potential theory of drizzle formation is estimated using the new results.