| Abstract: |
| During this talk, we present some results on the well-posedness of the martingale problem for non-linear stochastic differential equations in the sense of McKean-Vlasov under mild assumptions on the coefficients and a class of associated linear partial differential equations 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, that is, the space of probability measures on $\mathbb{R}^d$ with a finite second-order moment. The martingale problem is addressed by a perturbation argument on $\mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$, for non-linear coefficients including any bounded continuous drift and diffusion coefficient satisfying some structural assumption in the measure sense that covers a large class of interaction. Under additional assumptions, we then establish the existence and smoothness of the associated density as well as Gaussian type bounds, the derivatives with respect to the measure being understood in the sense introduced by P.-L. Lions. Finally, existence and uniqueness for the related linear Cauchy problem with irregular terminal condition and source term among the considered class of non-linear interaction is addressed. |
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