Strong well-posedness of McKean-Vlasov stochastic differential equation with Hölder drift

Paul-Éric Chaudru de Raynal (1)

1 LAMA - Laboratoire de Mathématiques

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.

Domaines: Probabilités [math.PR]

Paul-Éric Chaudru de Raynal. Strong well-posedness of McKean-Vlasov stochastic differential equation with Hölder drift. Stochastic Processes and their Applications, 2020, 130 (1), pp.79-107. ⟨10.1016/j.spa.2019.01.006⟩. ⟨hal-01248324v3⟩