Streaming process discovery aims to discover a process model that may change over time, coping with the challenges of concept drift in business processes. Existing studies update process models with fixed strategies, neglecting the highly dynamic nature of trace streams. Consequently, they fail to accurately reveal the process evolution caused by concept drift. This paper proposes RLSPD (Reinforcement Learning-based Streaming Process Discovery), a dynamic process discovery approach for constructing an online process model on a trace stream. RLSPD leverages conformance-checking information to characterize trace distribution and employs a reinforcement learning policy to capture fluctuations in the trace stream. Based on the dynamic parameters provided by reinforcement learning, we extract representative trace variants within a memory window using frequency-based sampling and perform concept drift detection. Upon detecting concept drift, the process model is updated by process discovery. Experimental results on real-life event logs demonstrate that our approach effectively adapts to the high dynamics of trace streams, improving the conformance of constructed process models to upcoming traces and reducing erroneous model updates. Additionally, the results highlight the significance of the pre-trained policy in dealing with unknown environments.