  We establish global well-posedness and convergence of the score-based
generative models (SGM) under minimal general assumptions of initial data for
score estimation. For the smooth case, we start from a Lipschitz bound of the
score function with optimal time length. The optimality is validated by an
example whose Lipschitz constant of scores is bounded at initial but blows up
in finite time. This necessitates the separation of time scales in conventional
bounds for non-log-concave distributions. In contrast, our follow up analysis
only relies on a local Lipschitz condition and is valid globally in time. This
leads to the convergence of numerical scheme without time separation. For the
non-smooth case, we show that the optimal Lipschitz bound is O(1/t) in the
point-wise sense for distributions supported on a compact, smooth and
low-dimensional manifold with boundary.
