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Selection problems for a discount degenerate viscous Hamilton-Jacobi equation. (English) Zbl 1357.35090
Summary: 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.

MSC:
35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
37J50 Action-minimizing orbits and measures (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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