Regularisation in mean field models via infinite dimensional diffusion

This talk aims to show how randomizing the dynamics in mean field models can help regularize the associated partial differential equations on the space of probability measures. A key challenge in this approach lies in constructing a suitable notion of noise on the space of probability measures. To this end, we rely on the Dirichlet–Ferguson diffusion process, as studied by Dello Schiavo [AOP 22]. We first examine the effect of this noise on a system of uncontrolled interacting particles and show that it induces a regularizing effect at the level of the corresponding backward Kolmogorov equation. We then analyze a mean field control problem driven by this noise and prove that the associated Hamilton–Jacobi equation admits a unique solution in an appropriate functional space, even when the coefficients have limited regularity. The talk is based on a joint work with F. Delarue (Nice) and G. Sodini (Vienna).