This paper investigates the smoothness properties of diffusion models, focusing on the case where the target data distribution is a mixture of Gaussians. Diffusion models have shown significant progress in generating high-quality samples across various domains. However, there is a lack of theoretical understanding regarding the Lipschitz continuity and second momentum properties of the diffusion process. The authors bridge this gap by analyzing these properties for the case of Gaussian mixtures, which are known to be universal approximators for smooth densities. The paper proves that if the target distribution is a k-mixture of Gaussians, the density of the entire diffusion process is also a k-mixture of Gaussians. This result allows for the derivation of tight upper bounds on the Lipschitz constant and second momentum, which are independent of the number of mixture components k. These bounds are then applied 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. The authors provide a detailed analysis of the Lipschitz and second momentum properties of Gaussian mixtures, leading to concrete bounds for the diffusion process. These results contribute to a deeper theoretical understanding of diffusion models and provide a foundation for further advancements in this field. The paper also discusses the implications of these findings for practical applications and highlights the importance of theoretical analysis in the development of diffusion models.This paper investigates the smoothness properties of diffusion models, focusing on the case where the target data distribution is a mixture of Gaussians. Diffusion models have shown significant progress in generating high-quality samples across various domains. However, there is a lack of theoretical understanding regarding the Lipschitz continuity and second momentum properties of the diffusion process. The authors bridge this gap by analyzing these properties for the case of Gaussian mixtures, which are known to be universal approximators for smooth densities. The paper proves that if the target distribution is a k-mixture of Gaussians, the density of the entire diffusion process is also a k-mixture of Gaussians. This result allows for the derivation of tight upper bounds on the Lipschitz constant and second momentum, which are independent of the number of mixture components k. These bounds are then applied 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. The authors provide a detailed analysis of the Lipschitz and second momentum properties of Gaussian mixtures, leading to concrete bounds for the diffusion process. These results contribute to a deeper theoretical understanding of diffusion models and provide a foundation for further advancements in this field. The paper also discusses the implications of these findings for practical applications and highlights the importance of theoretical analysis in the development of diffusion models.