Regularization by 1/2-Laplacian and vanishing viscosity approximation of HJB equations

TL;DR

This paper studies how adding a fractional Laplacian with exponent 1/2 to first order HJB equations enhances regularity and enables explicit convergence rates in vanishing viscosity approximations.

Contribution

It introduces regularity estimates for solutions with fractional Laplacian perturbations and derives explicit convergence rates for vanishing viscosity limits.

Findings

Regularity estimates for viscosity solutions with 1/2-Laplacian perturbation

Well-defined classical solutions for perturbed HJB equations

Explicit convergence rate for vanishing viscosity approximation

Abstract

We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 1/2, 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 1/2-Laplacian and derive an explicit rate convergence for the vanishing viscosity limit.