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

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.

💡 Research Summary

The paper investigates McKean‑Vlasov stochastic differential equations (SDEs) whose drift term is defined through a convolution with a highly singular interaction kernel belonging to a local negative Sobolev space (\tilde W^{-\delta,k}(\mathbb R^{d};\mathbb R^{d})). The kernel (h_{t}) may be a distribution rather than a classical function, allowing for interactions more singular than the usual Riesz kernels and even for drifts that depend on higher‑order derivatives of the density (Nemytskii‑type SDEs). The authors introduce two main methodological innovations. First, a “time‑shift” argument: the kernel is multiplied by a factor (t^{\kappa}) (with (\kappa\ge0)), which forces the singularity to vanish as (t\downarrow0). This technique relaxes the usual requirement that the interaction be bounded at the initial time and enables the treatment of kernels whose singularity would otherwise be prohibitive. Second, they develop a priori estimates based on the heat semigroup (P^{0}_{t}). Lemma 3.1 shows that for any (i=0,1,\infty) and suitable indices (\delta,\varepsilon,p,k) one has
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