Distribution Dependent SDEs with Singular Interactions: Well-Posedness and Regularity

Abstract: For a class of distribution dependent SDEs with singular interactions, which include the Coulomb/Riesz/Biot-Savart kernels as typical examples (Examples 2.1 and 2.2), we derive the well-posedness and regularity estimates by establishing the entropy-cost inequality. To measure the singularity of interactions, we introduce a new probability distance induced by local integrable functions, and estimate this distance for the time-marginal laws of solutions by using the Wasserstein distance of initial distributions. A key point of the study is to characterize the path space of time-marginal distributions for the solutions, by using local hyperbound estimates on diffusion semigroups.