Intrinsic regularization by noise for $1d$ mean field games

The purpose of this article is to show that an intrinsic noise with values in the space ${\mathcal P}({\mathbb R})$ of $1d$ probability measures may force uniqueness to first order mean field games. The structure of the noise is inspired from an earlier work [arXiv:2210.01239]. It reads as a coloured Ornstein-Uhlenbeck process with reflection on the boundary of quantile functions on the $1d$ torus, with the elements of the latter playing the role of indices for the continuum of players underpinning the game. In [arXiv:2210.01239], the semi-group generated by the noise is shown to enjoy smoothing properties that become key in the study carried out here. Although the analysis is limited to the 1d setting, this is the first example of uniqueness forcing for generic mean field games set over an infinite dimensional set of probability measures and this may be one step forward towards a more systematic regularization by noise theory for mean field games.