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

We study linear and nonlinear PDEs defined on the space of P(Tᵈ) over the flat torus Tᵈ, equipped with the Dirichlet-Ferguson measure D. We first develop an analytic framework based on the Wasserstein-Sobolev space H1,2(P(Tᵈ), W₂, 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.