Long-Time Relations between Mean-Field Games and Aggregation Equations

Definitions and mathematical statement

The supplied question concerns a mean-field-game system and an associated aggregation or kinetic equation. Neither the equations nor the meaning of their association are specified. Here, "long-time behavior" is retained as an unresolved term: the record does not specify whether it concerns evolution as time tends to infinity, a limit as a game horizon tends to infinity, or another asymptotic regime.

The source question is:

What is the relationship, if any, between the long-time behavior of mean-field-game systems and that of the associated aggregation or kinetic equation?

A precise quantified formulation requires the choice of systems and associated equations, spatial and state domains, coefficients and interaction or cost functions, solution concepts, admissible initial and boundary or terminal conditions, and the asymptotic quantities and convergence notions to be compared. These data are absent. In particular, the record does not determine whether a relationship is sought for every member of a specified class or only under additional hypotheses. No particular equations, hypotheses, or asymptotic correspondence are asserted here.

The supplied open-status assessment discusses a fast algorithm for branching Brownian optimal transport and a Sinkhorn-type complexity result, rather than the long-time comparison stated above. Its relevance to this question is unresolved.

Short English statement

The question asks whether the long-time behavior of mean-field games is related to that of an associated aggregation or kinetic equation. The models, their association, and the intended asymptotic comparison are unspecified, so the record does not support a precise theorem or conjecture. The supplied literature assessment concerns a different mathematical target.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004634$, source problem code catalogue entry, title "Long-Time Relations between Mean-Field Games and Aggregation Equations," in the collection "Oberwolfach Reports Open Problems." The recorded proposed year is 2021; no proposer is named.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2022), Oberwolfach Reports, pp. 507--587, DOI 10.4171/owr/2021/10. Supplied source URL and literature reference labeled "Oberwolfach Report 10/2021": https://doi.org/10.4171/owr/2021/10. The report is supplied as provenance for workshop open-problem discussions; its contents and the attribution of this particular question have not been independently verified here. The citation's 2022 title date differs from the 2021 report designation and recorded proposed year; their relationship is not explained in the record.
• Supplied literature assessment, checked on exactly 2026-08-22: status "open." The assessment says that the report identifies a fast algorithm for branching Brownian optimal transport as an open question and that no general Sinkhorn-type complexity result resolves it. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims. The assessment's connection to the present long-time mean-field-game question is unclear.
• The registry labels the statement corrected_verified and supplies a claim of manual semantic, typographic, and LaTeX review confirming self-containment and preservation of the mathematical target. These are supplied review labels, not independent verification of mathematical precision or correctness in this draft.