Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients

We develop a quantitative framework for the long-time behavior of McKean–Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit $L^1$-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean–Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie–Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.