Existence and Uniqueness for McKean-Vlasov equations with singular interactions

Abstract

We investigate the well-posedness of following McKean-Vlasov equation in $\mathbb{R}^d$: \[ \mathrm{d} X_t=\sigma(t,X_t, \mu_{X_t})\mathrm{d} W_t+b(t, X_t, \mu_{X_t}) \mathrm{d} t, \] where $\mu_{X_t}$ is the law of $X_t$. The existence of solutions is demonstrated when $\sigma$ satisfies certain non-degeneracy and continuity assumptions, and when $b$ meets some integrability conditions, and continuity requirements in the (generalized) total variation distance. Furthermore, uniqueness is established under additional continuity assumptions of a Lipschitz type.