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We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left( X(t), \displaystyle\frac{d\mathcal{L}_{X(t)}}{dx}(X(t))\right) dt+\sigma\left( X(t),\displaystyle\frac{d\mathcal{L}_{X(t)}}{dt}(X(t))\right) dW(t)$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$.
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Barbu, V., Röckner, M., Probabilistic representation for solutions to nonlinear Fokker-Planck equation, SIAM J. Math. Anal
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Wang, F.-Y., Distribution dependent SDEs for Landau type equations, Stochastic Process. Appl
2018
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