Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

TL;DR

This paper proves the convergence of solutions for a discounted degenerate viscous Hamilton--Jacobi equation to the ergodic problem, characterizing the limit via stochastic Mather measures using the nonlinear adjoint method.

Contribution

It extends convergence results to viscous Hamilton--Jacobi equations with convex Hamiltonians, employing the nonlinear adjoint method and a new commutation lemma.

Findings

Convergence of discounted solutions to the ergodic problem.

Characterization of the limit using stochastic Mather measures.

Extension of previous first-order results to viscous equations.

Abstract

We prove that the solution of the discounted approximation of a degenerate viscous Hamilton--Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by naturally using the nonlinear adjoint method, and deriving a commutation lemma. This convergence result was first achieved by Davini, Fathi, Iturriaga, and Zavidovique for the first order Hamilton--Jacobi equation.