Point cloud upsampling (PCU) enriches the representation of raw point clouds, significantly improving the performance in downstream tasks such as classification and reconstruction. Most of the existing point cloud upsampling methods focus on sparse point cloud feature extraction and upsampling module design. In a different way, we dive deeper into directly modelling the gradient of data distribution from dense point clouds. In this paper, we proposed a conditional denoising diffusion probabilistic model (DDPM) for point cloud upsampling, called PUDM. Specifically, PUDM treats the sparse point cloud as a condition, and iteratively learns the transformation relationship between the dense point cloud and the noise. Simultaneously, PUDM aligns with a dual mapping paradigm to further improve the discernment of point features. In this context, PUDM enables learning complex geometry details in the ground truth through the dominant features, while avoiding an additional upsampling module design. Furthermore, to generate high-quality arbitrary-scale point clouds during inference, PUDM exploits the prior knowledge of the scale between sparse point clouds and dense point clouds during training by parameterizing a rate factor. Moreover, PUDM exhibits strong noise robustness in experimental results. In the quantitative and qualitative evaluations on PU1K and PUGAN, PUDM significantly outperformed existing methods in terms of Chamfer Distance (CD) and Hausdorff Distance (HD), achieving state of the art (SOTA) performance.