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In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on the coefficients as well as classical solutions for a class of associated linear partial differential equations (PDEs) defined on $[0,T] \times \mathbb{R}^d \times \mathcal{P}\_2(\mathbb{R}^d)$, for any $T>0$, $\mathcal{P}\_2(\mathbb{R}^d)$ being the Wasserstein space (i.e.
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