Unique Nonequilibrium Invariant Measures for Resonant NLS

Definitions and mathematical statement

The supplied question concerns a resonant nonlinear Schrödinger (NLS) toy model coupled to heat baths whose temperatures satisfy $T_1<T_n$. The evolution equations, number of modes, state space, bath coupling, stochastic forcing, admissible parameters, and boundary conditions are not supplied. Consequently, the specific dynamics cannot be reconstructed from this record.

A conditional formulation of the requested property is as follows. Once the intended model is specified, let $(X,\mathcal B)$ be its measurable state space and let $(P_t)_{t\geq 0}$ be its Markov transition semigroup, if that is the intended stochastic formulation. A probability measure $\mu$ on $(X,\mathcal B)$ is invariant if \[ \int_X P_t f(x)\,\mu({\,\mathrm{d}} x)=\int_X f(x)\,\mu({\,\mathrm{d}} x) \quad\text{for every bounded measurable }f:X\to{\mathbb{R}}\text{ and every }t\geq 0. \] Under this formulation, an invariant probability measure is ergodic if it is an extreme point of the convex set of invariant probability measures: whenever $\mu=a\mu_1+(1-a)\mu_2$, with $0<a<1$ and invariant probability measures $\mu_1,\mu_2$, one has $\mu_1=\mu_2=\mu$.

Question, conditional on the missing model specifications. For the intended resonant NLS toy model with heat baths satisfying $T_1<T_n$, does there exist a unique smooth ergodic nonequilibrium invariant probability measure?

Here "smooth" is unresolved: the record specifies neither a differentiable structure nor a reference measure relative to which a density should be smooth, nor a required regularity class. The term "nonequilibrium" is retained from the source; the only explicit temperature condition is $T_1<T_n$, and no additional condition on currents or detailed balance is supplied. It is also unclear whether uniqueness is intended among all invariant probability measures or only among those satisfying the stated smoothness and ergodicity requirements. The record does not specify whether the question concerns one fixed parameter choice or every admissible choice satisfying the temperature inequality. These omissions prevent a fully specified mathematical statement.

Short English statement

The question asks whether a resonant NLS toy model connected to heat baths at unequal temperatures has a unique smooth ergodic stationary probability law. The model equations, parameter quantifiers, meaning of smoothness, and scope of uniqueness must be supplied before this becomes a precise problem. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003150$, source problem code catalogue entry, title "Unique Nonequilibrium Invariant Measures for Resonant NLS," in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied. The proposed year is recorded as $2016$.
• Supplied report citation: Nonlinear Evolution Problems (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/27, pp. 1517--1576. Supplied source URL: https://doi.org/10.4171/owr/2016/27. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2016/27. The exact supplied literature-check date is 2026-08-21. This assessment is supplied dated curation, not independently verified: it reports that no authoritative theorem establishing a unique smooth ergodic nonequilibrium invariant measure for the reported water-wave models was located, and labels the question open. The supplied background requests independent verification of time-sensitive status and cited claims. The relationship between those water-wave models and the resonant NLS toy model is not specified.
• The registry supplies the status open and the statement status corrected_verified; its verification note says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied registry qualifications, not independent mathematical or source verification of this draft.