Multidimensional stable driven McKean–Vlasov SDEs with distributional interaction kernel: a regularization by noise perspective

Abstract

In this work, we are interested in establishing weak and strong well-posedness for McKean–Vlasov SDEs with additive stable noise and a convolution type non-linear drift with singular interaction kernel in the framework of Lebesgue–Besov spaces. We prove that the well-posedness of the system holds for the thresholds (in terms of regularity indexes) deriving from the scaling of the noise and that the corresponding SDE can be understood in the classical sense. Especially, we characterize quantitatively how the non-linearity allows to go beyond the stronger thresholds, coming from Bony’s paraproduct rule, usually obtained for linear SDEs with singular interaction kernels. We also specifically characterize in function of the stability index of the driving noise and the parameters of the drift when the dichotomy between weak and strong uniqueness occurs.