The vanishing viscosity approximation for first-order MFGs with nonlocal coupling converges at rate O(ε^{1/2}) for the value function and O(ε^{1/8}) for the density in Wasserstein distance.
Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case
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abstract
We study the speed of convergence in $L^\infty$ norm of the vanishing viscosity process for Hamilton-Jacobi equations with uniformly or strictly convex Hamiltonian terms with superquadratic behavior. Our analysis boosts previous findings on the rate of convergence for this procedure in $L^p$ norms, showing rates in sup-norm of order $\mathcal{O}(\epsilon^\beta)$, $\beta\in(1/2,1)$, or $\mathcal{O}(\epsilon|\log\epsilon|)$ with respect to the vanishing viscosity parameter $\epsilon$, depending on the regularity of the initial datum of the problem and convexity properties of the Hamiltonian. Our proofs are based on integral methods and avoid the use of techniques based on stochastic control or the maximum principle.
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math.AP 2years
2026 2verdicts
ACCEPT 2representative citing papers
Vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the torus converge pointwise at rate O(ε|log ε|) and in averaged form at rate O(ε).
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On the rate of the vanishing viscosity approximation for Mean Field Games with nonlocal coupling
The vanishing viscosity approximation for first-order MFGs with nonlocal coupling converges at rate O(ε^{1/2}) for the value function and O(ε^{1/8}) for the density in Wasserstein distance.
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Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations
Vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the torus converge pointwise at rate O(ε|log ε|) and in averaged form at rate O(ε).