{"ID":2828597,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.14568","arxiv_id":"2512.14568","title":"On Viscosity Solutions of Hamilton-Jacobi Equations in the Wasserstein space and the Vanishing Viscosity Limit","abstract":"The aim of this article is twofold. First, we develop a unified framework for viscosity solutions to both first-order Hamilton-Jacobi equations and semilinear Hamilton-Jacobi equations driven by the idiosyncratic operator, defined on the Wasserstein Space. Second, we establish a vanishing-viscosity limit-extending beyond the classical control-theoretic setting-for solutions of semilinear Hamilton-Jacobi equations, proving their convergence to the corresponding first-order solution as the idiosyncratic noise vanishes. Our approach provides an optimal convergence rate. We also present some results of independent interest. These include existence theorems for the first-order equation, obtained through an appropriate Hopf-Lax representation, and a useful description of the action of the idiosyncratic operator on geodesically convex functions.","short_abstract":"The aim of this article is twofold. First, we develop a unified framework for viscosity solutions to both first-order Hamilton-Jacobi equations and semilinear Hamilton-Jacobi equations driven by the idiosyncratic operator, defined on the Wasserstein Space. Second, we establish a vanishing-viscosity limit-extending beyo...","url_abs":"https://arxiv.org/abs/2512.14568","url_pdf":"https://arxiv.org/pdf/2512.14568v2","authors":"[\"Giacomo Ceccherini Silberstein\",\"Daniela Tonon\"]","published":"2025-12-16T16:39:31Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.MG\",\"math.OC\"]","methods":"[]","has_code":false}