Well-posedness of density dependent SDE driven by $α$-stable process with Hölder drifts

Abstract: In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $\alpha$-stable processes with $\alpha \in(1,2)$. The existence part is based on Euler's approximation as \cite{HRZ20}, while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker-Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being $\beta_0$-order H\"older continuous in the spatial variable, where $\beta_0\in(1-\alpha/2,1)$.