As the volumes of spatiotemporal trajectory data continue to grow at a rapid pace; a new generation of data management techniques is needed in order to be able to utilize these data to provide a range of data-driven services, including geographic-type services. Key challenges posed by spatiotemporal data include the massive data volumes, the high velocity with which the data are captured, the need for interactive response times, and the inherent inaccuracy of the data. We propose an infrastructure, Elite, that leverages peer-to-peer and parallel computing techniques to address these challenges. The infrastructure offers efficient, parallel update and query processing by organizing the data into a layered index structure that is logically centralized, but physically distributed among computing nodes. The infrastructure is elastic with respect to storage, meaning that it adapts to fluctuations in the storage volume, and with respect to computation, meaning that the degree of parallelism can be adapted to best match the computational requirements. Further, the infrastructure offers advanced functionality, including probabilistic simulations, for contending with the inaccuracy of the underlying data in query processing. Extensive empirical studies offer insight into properties of the infrastructure and indicate that it meets its design goals, thus enabling the effective management of big spatiotemporal data.