Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations

We study quantitative convergence rates for vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the flat torus. The limiting equation contains a spatially dependent diffusion coefficient a(x) >= 0, which is allowed to vanish. Under standard structural assumptions on the Hamiltonian, we first prove a pointwise convergence rate of order O(epsilon |log epsilon|). We then show that, when the error is tested against a smooth probability density, the logarithmic loss can be removed and an averaged O(epsilon) rate holds. The proof is based on the nonlinear adjoint method, weighted Hessian estimates, and entropy estimates for the adjoint density.