Controllability and Exponential Mixing in Singular Interacting Particle Systems

This article concerns interacting particle systems with singular kernels, driven either by degenerate deterministic controls or by degenerate decomposable noise. In the deterministic setting, we establish global exact controllability and a topologically robust property called solid controllability. Moreover, we prove a result that guarantees global approximate controllability with prescribed trajectories. For stochastic dynamics, we obtain ergodicity and exponential mixing by utilizing coupling and recurrence mechanisms based on controllability. Our approach exploits the singularity and applies to a broad class of models, including Biot-Savart, Coulomb, Riesz, and Yukawa interactions, as well as heterogeneous multi-species systems.