WELL-POSEDNESS FOR SOME NON-LINEAR DIFFUSION PROCESSES AND RELATED PDE ON THE WASSERSTEIN SPACE - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2022

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 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 (\emph{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.
Fichier principal
Vignette du fichier
Non_Linear_Diffusion_and_Related_PDE.pdf (1.37 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01924949 , version 1 (16-11-2018)
hal-01924949 , version 2 (04-07-2019)
hal-01924949 , version 3 (20-04-2021)

Identifiants

Citer

Paul-Eric Chaudru de Raynal, Noufel Frikha. WELL-POSEDNESS FOR SOME NON-LINEAR DIFFUSION PROCESSES AND RELATED PDE ON THE WASSERSTEIN SPACE. Journal de Mathématiques Pures et Appliquées, 2022, ⟨10.1016/j.matpur.2021.12.001⟩. ⟨hal-01924949v3⟩
269 Consultations
192 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More