This paper presents the Stochastic Harmonic Potential Field (SHPF) method for provably safe navigation of differential-drive mobile robots in dynamic environments using 2D LiDAR sensing. SHPF combines harmonic potential fields, velocity-adaptive stochastic prediction of obstacle motion, and an adaptive control law to ensure local-minima-free trajectories with real-time safety guarantees. Theoretical analysis establishes sufficient conditions for collision avoidance under maximum velocity constraints, while extensive simulations demonstrate superior performance over five state-of-the-art methods (DWA, VO, ORCA, RRT*, MPC). Evaluated in two benchmark scenarios—dynamic corridor navigation and cross-traffic avoidance—SHPF achieved a 100% success rate, the highest observed safety margin (0.37 m), and sub-13 ms control loop execution on embedded hardware. These results confirm the method’s robustness for real-world deployment in cluttered, unpredictable environments typical of logistics and human-populated spaces.