Nondissipative Dynamics from Antisymmetric Large Deviations

Definitions and mathematical statement

The supplied record denotes by $L_F^{\mathrm{asym}}$ a time-antisymmetric large-deviation rate function. It does not define $F$, give a formula or domain for $L_F^{\mathrm{asym}}$, specify the underlying state space, or state how this function generates dynamics. Accordingly, this notation is retained as a source designation, without assigning it an additional mathematical definition.

Source-qualified conjectural target. Determine whether the dynamics generated by $L_F^{\mathrm{asym}}$ admits a variational structure that is a counterpart of gradient flow and describes purely nondissipative dynamics.

The record supplies no quantified class of admissible systems, initial or boundary conditions, definition of the proposed variational structure, or mathematical criterion for purely nondissipative dynamics. Thus the target above preserves the source's conjectural assertion, but cannot be made into a fully specified mathematical proposition from the supplied evidence alone. In particular, no energy, entropy, conservation identity, Hamiltonian, or Poisson structure is specified. The background's reference to a Hamiltonian interpretation does not supply such a structure or establish its equivalence to nondissipation. The designation "time-antisymmetric" also lacks a specified time-reversal operation or decomposition; no sign-change identity for $L_F^{\mathrm{asym}}$ is asserted here.

Short English statement

The source asks whether the dynamics associated with $L_F^{\mathrm{asym}}$ has a variational description analogous to gradient flow for purely nondissipative evolution. The underlying rate function, admissible systems, and meaning of nondissipation are unspecified, so this remains a source-qualified conjectural target requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004509$, source problem code catalogue entry, titled Nondissipative Dynamics from Antisymmetric Large Deviations, in the Oberwolfach Reports Open Problems collection. The record assigns proposed year $2020$ and supplies no named proposer.
• Supplied source citation: Variational Methods for Evolution (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/29, pp. 1391--1467. https://doi.org/10.4171/owr/2020/29. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise page or passage for this problem is supplied.
• Patterson--Renger--Sharma, Variational Structures Beyond Gradient Flows, listed in the supplied literature sources: https://publications.imp.fu-berlin.de/2564/1/2103.14384.pdf.
• Oberwolfach Reports 2020/29, listed in the supplied literature sources: https://ems.press/content/serial-article-files/46865.
• Literature check date: 2026-08-22. This is supplied dated curation derived from the OWR record, not an independently verified assessment. It labels the problem open and states that the abstract Hamiltonian or nondissipative interpretation remains conjectural, that later work proves it only in selected particle models, and that the general framework remains formal. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting self-containment and preservation of the mathematical target. These are supplied editorial claims; they do not resolve the missing mathematical definitions or constitute independent verification here.