  Diffusion models have made rapid progress in generating high-quality samples
across various domains. However, a theoretical understanding of the Lipschitz
continuity and second momentum properties of the diffusion process is still
lacking. In this paper, we bridge this gap by providing a detailed examination
of these smoothness properties for the case where the target data distribution
is a mixture of Gaussians, which serves as a universal approximator for smooth
densities such as image data. We prove that if the target distribution is a
$k$-mixture of Gaussians, the density of the entire diffusion process will also
be a $k$-mixture of Gaussians. We then derive tight upper bounds on the
Lipschitz constant and second momentum that are independent of the number of
mixture components $k$. Finally, we apply our analysis to various diffusion
solvers, both SDE and ODE based, to establish concrete error guarantees in
terms of the total variation distance and KL divergence between the target and
learned distributions. Our results provide deeper theoretical insights into the
dynamics of the diffusion process under common data distributions.
