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arXiv · 1809.02216

Well-posedness of distribution dependent SDEs with singular drifts

Michael Röckner·Xicheng Zhang
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Abstract

Consider the following distribution dependent SDE: $$ {\mathrm d} X_t=σ_t(X_t,μ_{X_t}){\mathrm d} W_t+b_t(X_t,μ_{X_t}){\mathrm d} t, $$ where $μ_{X_t}$ stands for the distribution of $X_t$. 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 \cite{Kr-Ro} to the distribution dependent case.

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Michael Röckner, Xicheng Zhang. 2019-10-29. Well-posedness of distribution dependent SDEs with singular drifts. https://arxiv.org/abs/1809.02216

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