{"paper":{"title":"A new selection problem for degenerate viscous Hamilton-Jacobi equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The nonlinear adjoint method establishes uniform convergence to a distinguished ergodic solution via combined discounted approximation and potential perturbation for degenerate viscous Hamilton-Jacobi equations.","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Qinbo Chen, Zhi-Xiang Zhu","submitted_at":"2026-05-13T04:53:32Z","abstract_excerpt":"We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Hamiltonians are convex and the equations satisfy the degeneracy and viscosity conditions that allow the nonlinear adjoint method to produce the uniform convergence and the explicit formula in terms of generalized Mather measures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A selection principle for viscosity solutions of degenerate viscous Hamilton-Jacobi equations is derived via nonlinear adjoint methods, yielding uniform convergence to any desired ergodic solution expressed through generalized Mather measures and the potential.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The nonlinear adjoint method establishes uniform convergence to a distinguished ergodic solution via combined discounted approximation and potential perturbation for degenerate viscous Hamilton-Jacobi equations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"cb6149cb584d2c7ee8768ceb06f8c13086a69fd1350eeefb4d80ea25abc94863"},"source":{"id":"2605.12996","kind":"arxiv","version":1},"verdict":{"id":"5e2e12de-69ee-437d-a29a-3342d1251887","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:42:44.490158Z","strongest_claim":"Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.","one_line_summary":"A selection principle for viscosity solutions of degenerate viscous Hamilton-Jacobi equations is derived via nonlinear adjoint methods, yielding uniform convergence to any desired ergodic solution expressed through generalized Mather measures and the potential.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Hamiltonians are convex and the equations satisfy the degeneracy and viscosity conditions that allow the nonlinear adjoint method to produce the uniform convergence and the explicit formula in terms of generalized Mather measures.","pith_extraction_headline":"The nonlinear adjoint method establishes uniform convergence to a distinguished ergodic solution via combined discounted approximation and potential perturbation for degenerate viscous Hamilton-Jacobi equations."},"references":{"count":58,"sample":[{"doi":"","year":1998,"title":"M. Arisawa and P.-L. Lions. On ergodic stochastic control.Comm. Partial Differential Equations, 23(11-12):2187–2217, 1998","work_id":"98074f1e-31af-480b-a832-860afdad3229","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Armstrong and Hung V","work_id":"f7067076-f4be-4b9c-9904-f50620bd43d6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"Systems & Control: Foundations & Applications","work_id":"17aafffd-d4da-4ef2-958e-224e562584e5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"F. Cagnetti, D. Gomes, and H. V. Tran. Aubry–Mather measures in the nonconvex setting.SIAM J. Math. Anal., 43(6):2601–2629, 2011","work_id":"a76ea4dd-8d2e-45e4-b5de-a24d9c6ffa41","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Filippo Cagnetti, Diogo Gomes, Hiroyoshi Mitake, and Hung V. Tran. A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians.Ann. Inst. H. Poincaré ","work_id":"8e81eb6a-1fac-4324-9eff-88714e09cc83","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":58,"snapshot_sha256":"f8752eb6dfb0e9619351027cb01335b25e2b6d021b5874451aa586ca72c34f28","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}