  We examine effects on primordial nucleosynthesis from a truly random spatial
distribution in the baryon-to-photon ratio ($\eta$). We generate stochastic
fluctuation spectra characterized by different spectral indices and
root-mean-square fluctuation amplitudes. For the first time we explicitly
calculate the effects of baryon diffusion on the nucleosynthesis yields of such
stochastic fluctuations. We also consider the collapse instability of
large-mass-scale inhomogeneities. Our results are generally applicable to any
primordial mechanism producing fluctuations in $\eta$ which can be
characterized by a spectral index. In particular, these results apply to
primordial isocurvature baryon fluctuation (PIB) models. The amplitudes of
scale-invariant baryon fluctuations are found to be severely constrained by
primordial nucleosynthesis. However, when the $\eta$ distribution is
characterized by decreasing fluctuation amplitudes with increasing length
scale, surprisingly large fluctuation amplitudes on the baryon diffusion scale
are allowed.
