This paper addresses the theoretical understanding of Lipschitz continuity and second momentum properties in diffusion models, particularly when the target data distribution is a mixture of Gaussians. The authors prove that if the target distribution is a $k$-mixture of Gaussians, the entire diffusion process will also be a $k$-mixture of Gaussians. They derive tight upper bounds on the Lipschitz constant and second momentum that are independent of the number of mixture components $k$. The analysis is applied to various diffusion solvers, both SDE and ODE-based, to establish concrete error guarantees in terms of total variation distance and KL divergence between the target and learned distributions. The results provide deeper insights into the dynamics of diffusion processes under common data distributions.This paper addresses the theoretical understanding of Lipschitz continuity and second momentum properties in diffusion models, particularly when the target data distribution is a mixture of Gaussians. The authors prove that if the target distribution is a $k$-mixture of Gaussians, the entire diffusion process will also be a $k$-mixture of Gaussians. They derive tight upper bounds on the Lipschitz constant and second momentum that are independent of the number of mixture components $k$. The analysis is applied to various diffusion solvers, both SDE and ODE-based, to establish concrete error guarantees in terms of total variation distance and KL divergence between the target and learned distributions. The results provide deeper insights into the dynamics of diffusion processes under common data distributions.