{"ID":2845830,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.03522","arxiv_id":"2511.03522","title":"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.","short_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 $H^{1,2}(\\mathcal{P}(\\mathbb{T}^d), W_2, \\mathcal{D})$ asso...","url_abs":"https://arxiv.org/abs/2511.03522","url_pdf":"https://arxiv.org/pdf/2511.03522v1","authors":"[\"François Delarue\",\"Mattia Martini\",\"Giacomo Enrico Sodini\"]","published":"2025-11-05T14:54:44Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.AP\",\"math.PR\"]","methods":"[\"Diffusion Model\"]","has_code":false}