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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.03522 |
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| _version_ | 1866917062692044800 |
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| author | Delarue, François Martini, Mattia Sodini, Giacomo Enrico |
| author_facet | Delarue, François Martini, Mattia Sodini, Giacomo Enrico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | HJB equations driven by the Dirichlet-Ferguson Laplacian in Wasserstein-Sobolev spaces Delarue, François Martini, Mattia Sodini, Giacomo Enrico Optimization and Control Analysis of PDEs Probability Primary: 35R15, 60H15, 49L25, 49N80, Secondary: 60J60, 35K90, 46E35 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. |
| title | HJB equations driven by the Dirichlet-Ferguson Laplacian in Wasserstein-Sobolev spaces |
| topic | Optimization and Control Analysis of PDEs Probability Primary: 35R15, 60H15, 49L25, 49N80, Secondary: 60J60, 35K90, 46E35 |
| url | https://arxiv.org/abs/2511.03522 |