Strong well posedness of McKean–Vlasov stochastic differential equations with Hölder drift

Abstract

Here, we prove strong well-posedness for stochastic systems of McKean–Vlasov type with Hölder drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. The Hölder regularity of the drift with respect to the law argument being for the Wasserstein distance.Our proof is based on Zvonkin’s transformation (1974) and so on the regularization properties of the associated PDE, which is stated on the space [0,T]×Rd×P2(Rd), where T>0, d denotes the dimension equation and P2(Rd) is the space of probability measures on Rd 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.