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

TL;DR

This paper develops an analytical framework for PDEs on the space of probability measures over a torus, using Wasserstein-Sobolev spaces and the Dirichlet-Ferguson measure, establishing existence, uniqueness, and connections to particle systems.

Contribution

It introduces a novel Wasserstein-Sobolev space framework for PDEs driven by the Dirichlet-Ferguson Laplacian, linking PDEs with particle systems and extending to control problems.

Findings

Established existence and uniqueness of solutions for transport-diffusion and Hamilton-Jacobi equations.

Connected PDE solutions with interacting particle system representations.

Extended the framework to semilinear equations and mean-field control problems.

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.