The Smoluchowski–Kramers approximation for a system with arbitrary friction depending on both state and distribution

In this paper, we study a class of stochastic differential equations modeling diffusive phenomena with state-dependent and distribution-dependent friction. In the small-mass regime, the dynamics are shown to be governed by the Smoluchowski–Kramers approximation. We obtain the limiting equation and characterize the resulting additional drift terms via solutions of associated Lyapunov and Sylvester matrix equations. Moreover, we establish the rate of convergence and extend the model to incorporate more general interaction mechanisms and noise structures.