Regularities for McKean-Vlasov SDEs with Local Distributional Interactions
Xing Huang, Panpan Ren, Feng-Yu Wang
TL;DR
The paper advances the theory of McKean-Vlasov SDEs with local distributional interactions by introducing a local distributional framework $\mathscr P_{\delta,k*}$ and a maximal $\mathscr C_{\varepsilon,p;\delta,k}$-solution concept, enabling well-posedness for drift kernels $h_t\in \tilde W^{-\delta,k}$ with arbitrary singularity. A time-shift argument complements a fixed-point approach to establish global existence and uniqueness under broad conditions, including highly singular kernels and density-derivative dependencies (Nemytskii-type SDEs). Furthermore, it provides sharp regularity results for time-marginal laws via entropy-cost inequalities and $\| abla^i P_t^*\|$-type bounds, linking evolution to the Wasserstein distance of initial data. These results extend well-posedness and regularity to interactions beyond Riesz kernels and broaden applicability to density-derivative and distribution-dependent dynamics, with potential implications for kinetic-type equations and singular stochastic systems.
Abstract
Let $ \tilde W^{-δ,k}$ be the local negative Sobolev space on $\mathbb{R}^d$ with indexes $δ\in [0,\infty)$ and $k\in [1,\infty].$ We study McKean-Vlasov SDEs with interaction kernels in $\tilde W^{-δ,k}.$ By developing a time-shift argument which allows the singular interactions vanishing at time $0$, the global well-posedness is proved for regular enough initial distributions and any singular indexes $(δ,k)\in [0,\infty)\times [1,\infty],$ and for any initial distributions provided $δ+\frac{d}{k} < 1$. Moreover, the relative entropy and the $\|\cdot\|_{δ,k*}$-distance induced by $\tilde W^{-δ,k}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which point-wisely depend on the density function and its derivatives, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels.