To privately publish sensitive multimedia data in an edge network with fog devices, one of the best privacy-preserving solutions is to use differential privacy (DP) mechanisms. However, existing DP data publication mechanisms for the infinite data stream of edge networks mainly focus on publishing data with specific types of data or a set of predetermined queries. This approach is not suitable for multimedia data with numerous features that require a more flexible data publishing mechanism. In this paper, we propose EdgeSyn, a novel mechanism for accurately publishing multimedia data over infinite data streams in an edge network. It allocates privacy budgets with a sliding window, adopting data synthesis mechanisms to support dynamic publishing without loss of accuracy. In more detail, EdgeSyn addresses the limitations associated with data types in prior data stream publishing approaches and introduces a privacy budget management strategy that optimally allocates budgets for the implementation of data synthesis mechanisms over an infinite data stream. The experimental results show that EdgeSyn performs well under different privacy budgets and various lengths of active windows.