WELL-POSEDNESS FOR SOME NON-LINEAR DIFFUSION PROCESSES AND RELATED PDE ON THE WASSERSTEIN SPACE
Résumé
In this paper, we investigate the well-posedness of the martingale problem for 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 ] × R d × P 2 (R d), for any T > 0, P 2 (R d) being the Wasserstein space, that is, the space of probability measures on R d with a finite second-order moment. The martingale problem is addressed by a perturbation argument on R d ×P 2 (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. Some new well-posedness results in the strong sense also directly stem from the previous analysis. 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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