Exponential Ergodicity for Singular McKean–Vlasov Stochastic Differential Equations in Weighted Variation Metric

In this paper, we prove exponential ergodicity for McKean–Vlasov stochastic differential equations (SDEs) with singular drift under a weighted variation metric. The McKean–Vlasov SDE is perturbed by a singular potential and does not completely satisfy the typical dissipative condition in the x-variable. Our conclusion extends some ergodicity results in total variation norm with or without dependence on distribution and indicates ergodicity under Wasserstein distance if the weight function is chosen in a particular way. Furthermore, we apply the main result to nonlinear Fokker–Planck equations, in particular, to non-symmetric singular granular media equations, and observe the long-time behavior of the SDEs.