Well-posedness of distribution dependent SDEs with singular drifts

Abstract

Consider the following distribution dependent SDE:

$$\mathrm{d}{\mathit{X}}_{\mathit{t}}={\mathit{\sigma }}_{\mathit{t}}({\mathit{X}}_{\mathit{t}},{\mathit{\mu }}_{{\mathit{X}}_{\mathit{t}}})\mathrm{d}{\mathit{W}}_{\mathit{t}}+{\mathit{b}}_{\mathit{t}}({\mathit{X}}_{\mathit{t}},{\mathit{\mu }}_{{\mathit{X}}_{\mathit{t}}})\mathrm{d}\mathit{t},$$

where ${\mathit{\mu }}_{{\mathit{X}}_{\mathit{t}}}$ stands for the distribution of ${\mathit{X}}_{\mathit{t}}$. In this paper for non-degenerate σ, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in μ about b and σ. In particular, we extend the results of Krylov–Röckner (Probab. Theory Related Fields 131 (2005) 154–196) to the distribution dependent case.