Stability Beyond Unstable Population-Process Fluid Limits

Definitions and mathematical statement

The source concerns a peer-to-peer population process for a file consisting of two chunks, with a protocol called Deterministic First Chunk. Write $X$ for this stochastic population process and $\Phi$ for its deterministic large-system-limit dynamical system. These are descriptive labels: the supplied record does not specify the state space, transition rules, protocol mechanics, parameter range, initial conditions, or limiting procedure.

Source conjecture. Determine whether the stochastic Deterministic First Chunk process $X$ is stable, as suggested by the simulations reported in the supplied account, despite the reported instability of $\Phi$. Prove or disprove this stochastic stability conjecture for the intended model and parameter regime. An affirmative result, together with the claimed instability of the corresponding limit, would exhibit a population process whose instability does not follow from instability of its large-system limit.

The intended conjunction can only be recorded schematically as \[ \mathsf{Stable}_{\mathrm{stochastic}}(X) \quad\text{and}\quad \mathsf{Unstable}_{\mathrm{limit}}(\Phi). \] Neither predicate is defined by the supplied record. In particular, it does not specify whether stochastic stability means positive recurrence, existence of an invariant probability distribution, tightness, or another property; nor does it specify the deterministic instability criterion or the equilibrium, trajectory, or invariant set to which that criterion applies. These alternatives are unresolved possibilities, not adopted definitions. The parameter and initial-condition quantifiers are also absent. Consequently, the displayed conjunction is a statement schema requiring the original model definitions before it becomes a precise mathematical proposition.

The supplied dated assessment attributes instability of the large-system limit to later work and describes stochastic stability as a simulation-supported conjecture. Neither that instability assertion nor the supplied open status is independently verified here. Disproving stochastic stability for this particular process would not, by itself, settle whether some other population process can be stable while its large-system limit is unstable.

Short English statement

The supplied conjecture asks whether the two-chunk Deterministic First Chunk population process is stochastically stable even though its deterministic large-system limit is reported to be unstable. The process rules, stability definitions, parameter range, and required quantifiers are missing, so a fully precise formulation requires source clarification. The supplied open status and literature assessment remain dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001608$, title Stability Beyond Unstable Population-Process Fluid Limits, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2010$ and supplies no named proposer.
• Mathematical Challenges in Stochastic Networks (2011), Oberwolfach Reports, pp. 2745--2812, DOI 10.4171/owr/2010/48. Supplied source report and literature-source label: "original problem report." https://doi.org/10.4171/owr/2010/48. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source label: "Norros Reittu and Eirola stability paper." https://doi.org/10.1007/s11134-011-9209-2. The record supplies no full bibliographic entry or theorem locator for this reference.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem open, describes stochastic stability as a simulation-supported conjecture in the report, and states that later work proves instability of the large-system limit without settling the stochastic claim in this record. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement "exact" and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata; no source was visited or independently checked for this draft.