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

Abstract

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 \([0,T]\times \R^d\times \mathcal{P}\_2(\R^d)\), where \(T\) is a positive number, \(d\) denotes the dimension equation and \(\mathcal{P}\_2(\R^d)\) is the space of probability measures on \(\R^d\) 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.