Well-Posedness for Some Non-Linear Diffusion Processes and Related PDE on the Wasserstein Space

TL;DR

This paper establishes well-posedness and regularity results for non-linear diffusion processes and related PDEs on the Wasserstein space, using fixed point methods and Lions' derivatives, with implications for densities and linear PDEs.

Contribution

It provides new well-posedness results for McKean-Vlasov SDEs and associated PDEs on Wasserstein space, including strong solutions and density regularity under mild assumptions.

Findings

Proved existence and uniqueness of solutions to non-linear SDEs on Wasserstein space.

Derived Gaussian bounds for the derivatives of the density.

Addressed linear PDEs with irregular terminal conditions and source terms.

Abstract

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.