HJB equations driven by the Dirichlet-Ferguson Laplacian in Wasserstein-Sobolev spaces

Abstract

We study linear and nonlinear PDEs defined on the space of $\mathcal{P}(\mathbb{T}^d)$ over the flat torus $\mathbb{T}^d$, equipped with the Dirichlet-Ferguson measure $\mathcal{D}$. We first develop an analytic framework based on the Wasserstein-Sobolev space $H^{1,2}(\mathcal{P}(\mathbb{T}^d), W_2, \mathcal{D})$ associated with the Dirichlet form induced by the infinite-dimensional Laplacian acting on functions of measures. Within this setting, we establish existence and uniqueness results for transport-diffusion and Hamilton-Jacobi equations in the Wasserstein space. Our analysis connects the PDE approach with a corresponding interacting particles system providing a probabilistic (Kolmogorov-type) representation of strong solutions. Finally, we extend the theory to semilinear equations and mean-field optimal control problems, together with consistent finite-dimensional approximations.