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arXiv · 2605.22016

First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs

Jianbo Cui·Tonghe Dang
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Abstract

We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.

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Jianbo Cui, Tonghe Dang. 2026-05-21. First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs. https://arxiv.org/abs/2605.22016

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