With the proliferation of location-aware devices, large amount of trajectories have been generated when agents such as people, vehicles and goods flow around the urban environment. These raw trajectories, typically collected from various sources such as GPS in cars, personal mobile devices, and public transport, are often sparse and fragmented due to limited sampling rates, infrastructure coverage and data loss. In this context, trajectory recovery aims to reconstruct such sparse raw trajectories into their dense and continuous counterparts, so that fine-grained movement of agents across space and time can be captured faithfully. Existing trajectory recovery approaches typically rely on the prior knowledge of travel mode or motion patterns, and often fail in densely populated urban areas where accurate maps are absent. In this paper, we present a new recovery framework called TrajWeaver based on probabilistic diffusion models, which is able to recover dense and refined trajectories from the sparse raw ones, conditioned on various auxiliary features such as Areas of Interest along the way, user identity and waybill information. The core of TrajWeaver is a novel State Propagation Diffusion Model (SPDM), which introduces a new state propagation mechanism on top of the standard diffusion models, so that knowledge computed in earlier diffusion steps can be reused later, improving the recovery performance while reducing the number of steps needed. Extensive experiments show that the proposed TrajWeaver can recover from raw trajectories of various lengths, sparsity levels and heterogeneous travel modes, and outperform the state-of-the-art baselines significantly in recovery accuracy. Our code is available at: https://anonymous.4open.science/r/TrajWeaver/