Well-posedness of distribution dependent SDEs with singular drifts

Abstract

Consider the following distribution dependent SDE: $$\mathrm{d}{X}_t={\sigma }_t(X_t,{\mu }_{X_t})\mathrm{d}{W}_t+b_t(X_t,{\mu }_{X_t})\mathrm{d}t,$$ where ${\mu }_{X_t}$ stands for the distribution of $X_t$. In this paper for non-degenerate $\sigma$, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in $\mu$ about $b$ and $\sigma$. In particular, we extend the results of Krylov-Röckner \cite{Kr-Ro} to the distribution dependent case.

25 pages

References in corpus (4)

• Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
• Heat kernel and ergodicity of SDEs with distributional drifts
• Ergodicity of stochastic differential equations with jumps and singular coefficients
• A discretized version of Krylov's estimate and its applications

Cited by in corpus (1)

• Distribution dependent SDEs driven by additive continuous noise