A large spectrum of applications such as location based services and environmental monitoring demand efficient query processing on uncertain databases. In this paper, we propose the probabilistic Voronoi diagram (PVD) for processing moving nearest neighbor queries on uncertain data, namely the probabilistic moving nearest neighbor (PMNN) queries. A PMNN query finds the most probable nearest neighbor of a moving query point continuously. To process PMNN queries efficiently, we provide two techniques: a pre-computation approach and an incremental approach. In the pre-computation approach, we develop an algorithm to efficiently evaluate PMNN queries based on the pre-computed PVD for the entire data set. In the incremental approach, we propose an incremental probabilistic safe region based technique that does not require to pre-compute the whole PVD to answer the PMNN query. In this incremental approach, we exploit the knowledge for a known region to compute the lower bound of the probability of an object being the nearest neighbor. Experimental results show that our approaches significantly outperform a sampling based approach by orders of magnitude in terms of I/O, query processing time, and communication overheads.