Extrinsic derivative formula for distribution dependent SDEs

To characterize the regularity of nonlinear Fokker-Planck equations with respect to weighted variational distances, we establish for the first time a Bismut type formula for the extrinsic derivative of distribution dependent SDEs (DDSDEs). As an application, the Lipschitz continuity in the weighted variational distance is derived for the associated nonlinear Fokker-Planck equation, which can be regarded as the counterpart of the classical contraction property in the linear setting. The main results are illustrated by non-degenerate DDSDEs with space-time singular drift, as well as degenerate DDSDEs with weakly monotone coefficients. © 2025 ISI/BS.