Equilibration from Log-Sobolev Inequalities

Definitions and mathematical statement

The supplied record does not determine a standalone open problem. Its heading asks, "How fast does it equilibrate?", but no dynamics or particular equilibrium measure is specified. The record's verification note states that this question introduces log-Sobolev inequalities and quantitative equilibration bounds in the report.

The state space in the excerpt is $X={\mathbb{R}}^\Lambda$, with the index set $\Lambda$ unspecified. For a probability measure $\nu$ on $X$ and a positive measurable function $G$ with finite positive integral, define \[ \operatorname{Ent}_\nu(G)=\int_X G\log\!\left(\frac{G}{\int_XG\,\mathrm d\nu}\right)\,\mathrm d\nu. \] For probability measures $\mu\ll\nu$, with density $f=\mathrm d\mu/\mathrm d\nu$, define relative entropy by \[ H(\mu\mid\nu)=\int_X f\log f\,\mathrm d\nu =\operatorname{Ent}_\nu(f), \] with $0\log0=0$ and the value $+\infty$ permitted. Define the total variation norm of the difference of two probability measures by \[ \|\mu-\nu\|_{\mathrm{TV}}=2\sup_{A\text{ measurable}}|\mu(A)-\nu(A)|. \] This convention makes the displayed Pinsker bound below consistent; the excerpt does not specify its convention.

The first displayed formula in the supplied extraction suggests the log-Sobolev normalization \[ \operatorname{Ent}_\nu(G)\leq\frac{1}{2\gamma}I_\nu(G),\qquad \gamma>0, \] for every bounded smooth strictly positive function $G:{\mathbb{R}}^\Lambda\to(0,\infty)$. The excerpt calls the largest admissible $\gamma$ the log-Sobolev constant. However, its extracted definition has displaced factors and does not unambiguously preserve the coefficient. Moreover, neither $I_\nu$ nor the meaning of smoothness on ${\mathbb{R}}^\Lambda$ is specified. For finite $\Lambda$, a conventional Euclidean interpretation would be \[ I_\nu(G)=\int_{{\mathbb{R}}^\Lambda}\frac{|\nabla G|^2}{G}\,\mathrm d\nu, \] but this interpretation is conditional and is not recoverable as an explicit definition from the supplied text.

Writing $\nu_t$ for the unspecified time-dependent law and $\nu_\infty$ for its intended equilibrium, the excerpt asserts \[ H(\nu_t\mid\nu_\infty) \leq\frac{1}{2\gamma}I(\nu_t\mid\nu_\infty) =\frac{1}{2\gamma}I_{\nu_\infty}\!\left(\frac{\mathrm d\nu_t}{\mathrm d\nu_\infty}\right) \] and \[ \|\nu_t-\nu_\infty\|_{\mathrm{TV}}^2 \leq2H(\nu_t\mid\nu_\infty) \leq2e^{-2\gamma t}H(\nu_0\mid\nu_\infty), \qquad t\geq0. \] The first inequality in the second display is attributed to Pinsker. The excerpt describes equilibration as having rate $\gamma$; its display has exponent $2\gamma$ for entropy and squared total variation. Absolute continuity is needed for the displayed densities, and finite initial entropy is needed for a finite bound. Deriving time decay also requires a specified evolution and an appropriate entropy-dissipation relation with compatible normalization; these hypotheses are absent. Consequently, these displays preserve the asserted quantitative content but do not constitute a fully specified theorem or a recoverable research question. No model-specific rate or sharp-constant problem can be stated without additional source information.

Short English statement

The excerpt presents log-Sobolev inequalities as a route to exponential equilibration, with an asserted entropy bound proportional to $e^{-2\gamma t}$. It does not identify the dynamics, equilibrium measure, or a precise unresolved target. The supplied verification note identifies the passage as introductory exposition, although the registry labels it open. The extracted normalization and missing analytic hypotheses require human review.

Sources

• Local UnsolvedMath registry, permanent ID $30005722$, title: "Equilibration from Log-Sobolev Inequalities"; exact source problem code: catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2023.
• Supplied report citation: Arbeitsgemeinschaft: QFT and Stochastic PDEs (2024), Oberwolfach Reports, pp. 3319--3370, DOI 10.4171/owr/2023/59. Supplied source and literature link, labeled "Oberwolfach Report 59/2023": https://doi.org/10.4171/owr/2023/59. The citation's 2024 designation and the report's 2023 designation are retained as supplied.
• The extracted passage names Henri Elad Altman and Chunqiu Song. Their precise role is not specified, and the registry supplies no proposer attribution.
• Literature assessment checked at 2026-08-22: the supplied curation labels the item open and says that log-Sobolev theory gives exponential equilibration under a stated inequality, while the extracted question specifies neither a model nor a sharp rate. This is supplied dated curation, not independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is unrecoverable. Its verification note says that the official report uses the question to introduce log-Sobolev inequalities and then supplies quantitative exponential equilibration bounds, rather than posing a standalone unresolved problem. This note is supplied evidence; the report has not been independently consulted for this draft.