[Paper Review] Selection problems for a discounted degenerate viscous Hamilton--Jacobi equation

Hiroyoshi Mitake, Hung V. Tran|arXiv (Cornell University)|Aug 13, 2014

Quantum chaos and dynamical systems20 references4 citations

TL;DR

This paper establishes the uniform convergence of solutions to the discounted degenerate viscous Hamilton--Jacobi equation to a unique solution of the ergodic problem as the discount parameter ε→0. Using the nonlinear adjoint method and a novel commutation lemma, the authors characterize the limit solution via stochastic Mather measures, extending previous results from first-order to viscous equations with degenerate diffusion.

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.

Motivation & Objective

• To establish the convergence of the solution $ u^{000110} $ of the discounted degenerate viscous Hamilton--Jacobi equation to a solution of the ergodic problem as $ \varepsilon \to 0 $.
• To characterize the limit solution in terms of stochastic Mather measures, extending the dynamical systems approach to viscous equations.
• To develop a commutation lemma and nonlinear adjoint method tailored for degenerate viscous HJ equations.
• To prove the convergence holds for the full sequence $ \varepsilon \to 0 $, not just subsequences, under convex Hamiltonian and degenerate diffusion assumptions.
• To generalize the selection problem results from first-order to viscous HJ equations with $ a(x) \geq 0 $, including the case $ a \equiv 0 $.

Proposed method

• Apply the nonlinear adjoint method to derive a commutation lemma that links viscosity solutions and distributional solutions.
• Construct stochastic Mather measures for the degenerate diffusion $ a(x)\Delta u $ using the dynamics of the associated stochastic control problem.
• Use the commutation lemma to show that viscosity subsolutions are almost everywhere subsolutions, enabling regularity and convergence analysis.
• Establish uniform convergence of $ u^{000110} $ to the ergodic solution $ u^0 $ by comparing $ u^{000110} $ with test functions in the set $ \mathcal{E} $, defined as solutions satisfying a certain integral condition.
• Prove convergence via subsequence analysis: show $ \liminf_{\varepsilon \to 0} u^\varepsilon(x) \geq u^0(x) $ and $ \limsup_{\varepsilon \to 0} u^\varepsilon(x) \leq u^0(x) $, implying uniform convergence.
• Use mollification and $ \eta $-regularization of test functions to handle the degeneracy of $ a(x) $, proving convergence rate $ \eta^{1/2} $ for error terms.

Experimental results

Research questions

• RQ1 Does the solution $ u^\varepsilon $ of the discounted degenerate viscous Hamilton--Jacobi equation converge uniformly to a solution of the ergodic problem as $ \varepsilon \to 0 $?
• RQ2 Can the limit solution be characterized in terms of stochastic Mather measures for degenerate diffusion?
• RQ3 How can the nonlinear adjoint method be adapted to viscous HJ equations with degenerate diffusion?
• RQ4 Is the convergence of $ u^\varepsilon $ to the ergodic solution valid for the full sequence $ \varepsilon \to 0 $, not just subsequences?
• RQ5 What is the role of the commutation lemma in connecting viscosity solutions with distributional solutions in the degenerate case?

Key findings

• The solution $ u^\varepsilon $ of the discounted equation $ \varepsilon u^\varepsilon + H(x, Du^\varepsilon) = a(x)\Delta u^\varepsilon $ converges uniformly to a solution $ u^0 $ of the ergodic problem $ H(x, Du^0) = a(x)\Delta u^0 $ as $ \varepsilon \to 0 $.
• The limit solution $ u^0 $ is uniquely characterized as the supremum of all functions $ \phi \in \mathcal{E} $ satisfying $ \phi(x) \leq \int_{\mathbb{T}^n \times \mathbb{R}^n} \phi(x) \, d\nu(x,p) $, where $ \nu $ is a stochastic Mather measure.
• A new commutation lemma is established, showing that viscosity subsolutions of the viscous HJ equation are almost everywhere subsolutions, which is crucial for the convergence proof.
• The convergence rate of the regularized error terms $ R_2^\eta $ is shown to be $ \eta^{1/2} $, matching the convergence rate of the mollified functions $ S^\eta $.
• The proof extends the selection problem result from first-order HJ equations (where $ a \equiv 0 $) to viscous HJ equations with degenerate diffusion, providing a unified framework via stochastic Mather measures.
• The limit solution $ u^0 $ is independent of the choice of subsequence, proving that the full sequence $ u^\varepsilon $ converges uniformly to $ u^0 $ as $ \varepsilon \to 0 $.