Computable Coherent Sets in Stochastic Dynamics

Definitions and mathematical statement

For notation only, if a time-dependent stochastic dynamical system has measurable state spaces $(E_t,\mathcal{E}_t)$ indexed by a time set $I$, a family of measurable sets is a collection \[ (A_t)_{t\in I},\qquad A_t\in\mathcal{E}_t\quad(t\in I). \] This notation does not specify the stochastic dynamics or define coherence; neither is specified in the supplied record.

Research question. Can coherent families of sets for a time-dependent stochastic dynamical system be characterized in a way that permits numerical computation, and can the resulting problem be solved efficiently in higher dimensions?

Unresolved formulation. The source does not specify the class of stochastic systems, the state spaces, the time set or horizon, or the defining criterion for coherence. In particular, it supplies no reference measures, probability-retention criterion, admissibility or nontriviality conditions on the sets, or approximation tolerances. It also leaves unspecified the computational input model, the meaning of dimension, and the criterion for efficiency. Consequently, the question is retained as a research objective: a quantified characterization or complexity assertion cannot be stated precisely from this record alone without adding hypotheses.

Short English statement

The question asks for a characterization of coherent families in time-dependent stochastic dynamics that enables numerical computation, together with efficient methods in higher dimensions. The meanings of coherence, the admissible systems, and computational efficiency remain unspecified and require human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30004683$, title Computable Coherent Sets in Stochastic Dynamics, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The registry labels the statement as exact and the problem as open; these are supplied metadata, not independently verified conclusions.
• Supplied report citation: Mini-Workshop: Analysis of Data-driven Optimal Control (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/23, pp. 1209--1257. https://doi.org/10.4171/owr/2021/23. The literature-source label is "Oberwolfach Report 23/2021." The record gives proposed year 2021, while the supplied citation title contains 2022; this discrepancy is unresolved.
• Supplied literature reference: Denner, Junge and Matthes, "computing coherent sets." https://doi.org/10.3934/jcd.2016008. This is the supplied descriptive label; full bibliographic details are not provided.
• Supplied literature reference: Froyland, Koltai and Stahn, "finite-time coherent sets." https://doi.org/10.1137/19M1261791. This is the supplied descriptive label; full bibliographic details are not provided.
• Supplied literature assessment, checked 2026-08-22: the recorded status is open. The assessment states that spectral and transfer-operator methods compute coherent families in important finite-dimensional settings, but that a general characterization supporting efficient computation in higher dimensions remains unavailable. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; its time-sensitive status and cited claims require independent verification.