  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. the space of probability measures on
$\mathbb{R}^d$ with a finite second-order moment). In this case, the derivative
of a map along a probability measure is understood in the Lions' sense. The
martingale problem is addressed by a fixed point argument on a suitable
complete metric space, under some mild regularity assumptions on the
coefficients that covers a large class of interaction. Also, new well-posedness
results in the strong sense are obtained from the previous analysis. Under
additional assumptions, we then prove the existence of the associated density
and investigate its smoothness property. In particular, we establish some
Gaussian type bounds for its derivatives. We eventually address the existence
and uniqueness for the related linear Cauchy problem with irregular terminal
condition and source term.
