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 using the nonlinear adjoint method and deriving a commutation lemma. This convergence result was first proven by Davini, Fathi, Iturriaga, and Zavidovique for first order Hamilton–Jacobi equations.
Mitake, H., & Tran, H. V. (2017). Selection problems for a discount degenerate viscous Hamilton–Jacobi equation. Advances in Mathematics, 306, 684–703. https://doi.org/10.1016/j.aim.2016.10.032