FROM THE BACKWARD KOLMOGOROV PDE ON THE WASSERSTEIN SPACE TO PROPAGATION OF CHAOS FOR MCKEAN-VLASOV SDES

Noufel Frikha (1) , Paul-Eric Chaudru de Raynal (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 LAMA - Laboratoire de Mathématiques

In this article, we provide some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is also established. Our analysis relies on the well-posedness of classical solutions to the backward Kolmogorov partial differential equations (PDEs) defined on the strip [0, T ] × R d × P 2 (R d), P 2 (R d) being the Wasserstein space, that is, the space of probability measures on R d with a finite second-order moment and also on the existence and uniqueness of a fundamental solution for the related parabolic linear operator here stated on [0, T ] × P 2 (R d).

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Noufel Frikha, Paul-Eric Chaudru de Raynal. FROM THE BACKWARD KOLMOGOROV PDE ON THE WASSERSTEIN SPACE TO PROPAGATION OF CHAOS FOR MCKEAN-VLASOV SDES. 2019. ⟨hal-02168772v1⟩