Strong well-posedness of McKean-Vlasov stochastic differential equation with H{\"o}lder drift

In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin’s transformation \cite{zvonkin_transformation_1974} and so on the regularization properties of the associated PDE, which is stated on the space $[0,T]\times {\mathbb{R}}^d\times \mathcal{P}\mathrm{\_}2({\mathbb{R}}^d)$, where $T$ is a positive number, $d$ denotes the dimension equation and $\mathcal{P}\mathrm{\_}2({\mathbb{R}}^d)$ is the space of probability measures on ${\mathbb{R}}^d$ with finite second order moment. Especially, a smoothing effect in the measure direction is exhibited. Our approach is based on a parametrix expansion of the transition density of the McKean-Vlasov process.