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pith:24MTSQK5

pith:2026:24MTSQK5ODS2V5MPTUVUGH4HBV

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A new selection problem for degenerate viscous Hamilton-Jacobi equations

Qinbo Chen, Zhi-Xiang Zhu

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.

arxiv:2605.12996 v1 · 2026-05-13 · math.AP · math.DS

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

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[1] M. Arisawa and P.-L. Lions. On ergodic stochastic control.Comm. Partial Differential Equations, 23(11-12):2187–2217, 1998 1998

[2] Armstrong and Hung V 2015

[3] Systems & Control: Foundations & Applications 1997

[4] F. Cagnetti, D. Gomes, and H. V. Tran. Aubry–Mather measures in the nonconvex setting.SIAM J. Math. Anal., 43(6):2601–2629, 2011 2011

[5] 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 2015

Receipt and verification

First computed	2026-05-18T03:09:00.487003Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d71939415d70e5aaf58f9d2b431f870d68a847620c8c4b9fb0e0d31ebb7c1775

Aliases

arxiv: 2605.12996 · arxiv_version: 2605.12996v1 · doi: 10.48550/arxiv.2605.12996 · pith_short_12: 24MTSQK5ODS2 · pith_short_16: 24MTSQK5ODS2V5MP · pith_short_8: 24MTSQK5

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/24MTSQK5ODS2V5MPTUVUGH4HBV \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d71939415d70e5aaf58f9d2b431f870d68a847620c8c4b9fb0e0d31ebb7c1775

Canonical record JSON

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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-05-13T04:53:32Z",
    "title_canon_sha256": "ab2df1b64c17eb9f09e949d5ddd7595ccefe0063521a92a2b243ddbb169a9436"
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