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

TL;DR

This paper establishes the first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over finite graphs, overcoming boundary degeneracy challenges.

Contribution

It introduces a weighted $L^1$ framework and analyzes a new geometric drift term to prove convergence, addressing boundary degeneracy issues in Wasserstein space schemes.

Findings

Proves first-order convergence of the schemes.

Develops a weighted $L^1$ approach with boundary-vanishing weights.

Provides uniform bounds for the weighted adjoint variable.

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.