Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations
Louis-Pierre Chaintron, Samuel Daudin
TL;DR
This work determines the precise vanishing-viscosity rate for the first-order Hamilton–Jacobi–Bellman equation with a purely quadratic Hamiltonian and Lipschitz terminal data, proving the optimal rate is $O(\varepsilon \log \varepsilon)$ and furnishing a matching lower bound. The authors combine a regularization by sup-convolution with entropy-based estimates along a nonlinear adjoint flow and leverage semiconcavity to control the Laplacian, yielding a novel integrated Laplacian bound that drives the rate. The results generalize beyond one dimension, extend to cases where $f$ may vanish, and are shown to be sharp via explicit constructions; in particular, they provide the first global lower bound in all dimensions and connect to mean-field and large deviation frameworks. Overall, the paper improves the classical $O(\sqrt{\varepsilon})$ benchmark, clarifies the role of semiconcavity, and introduces robust techniques that may inform related vanishing-noise and mean-field analyses.
Abstract
The purpose of this note is to provide an optimal rate of convergence in the vanishing viscosity regime for first-order Hamilton-Jacobi equations with purely quadratic Hamiltonian. We show that for a globally Lipschitz-continuous terminal condition the rate is of order O($ε$ log $ε$), and we provide an example to show that this rate cannot be sharpened. This improves on the previously known rate of convergence O( $\sqrt$ $ε$), which was widely believed to be optimal. Our proof combines techniques involving regularization by sup-convolution with entropy estimates for the flow of a suitable version of the adjoint linearized equation. The key technical point is an integrated estimate of the Laplacian of the solution against this flow. Moreover, we exploit the semiconcavity generated by the equation.