Regularization by 12-Laplacian and vanishing viscosity approximation of HJB equations

We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 12, to a general first order HJB equation. Our results include some regularity estimates for the viscosity solutions of such perturbations, making the solutions classically well-defined. Most importantly, we use these regularity estimates to study the vanishing viscosity approximation to first order HJB equations by 12-Laplacian and derive an explicit rate convergence for the vanishing viscosity limit.