McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates

TL;DR

This paper establishes well-posedness and entropy-cost estimates for McKean-Vlasov SDEs with highly singular local distributional interactions in negative Sobolev spaces, extending the class of solvable models.

Contribution

It introduces a novel time-shift method to prove global well-posedness for McKean-Vlasov SDEs with singular interaction kernels in negative Sobolev spaces.

Findings

Proved global well-posedness under broad conditions.

Derived entropy and distance estimates for solution distributions.

Applicable to SDEs with highly singular interactions like Riesz kernels.

Abstract

Let $ \tt W^{-\dd,k}$ be the local negative Sobolev space on $\R^d$ with indexes $\dd \in [0,\infty)$ and $k\in [1,\infty].$ We study McKean-Vlasov SDEs with interaction kernels in $\tt W^{-\dd,k}.$ 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 $(\dd,k)\in [0,\infty)\times [1,\infty],$ and for any initial distributions provided $\dd+\ff d k<1$. Moreover, the relative entropy and the $\|\cdot\|_{\dd,k*}$-distance induced by $ \tt W^{-\dd,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 depending on higher order derivatives of density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels.