Weak solutions of McKean-Vlasov SDEs with supercritical drifts

Abstract

Consider the following McKean-Vlasov SDE: $$d{X}_t=\sqrt{2}d{W}_t+{\int }_{{\mathbb{R}}^d}K(t,X_t-y){\mu }_{X_t}(dy)dt,X_0=x,$$ where ${\mu }_{X_t}$ stands for the distribution of $X_t$ and $K(t,x):{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L_t^q({\tilde{L}}_x^p)$ with $\frac{d}{p}+\frac{2}{q}<2$, where ${\tilde{L}}_x^p$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations with measure as initial vorticity.

arXiv comment11pages