Low-density lineage limit of coalescing branching random walk

Definitions and mathematical statement

Consider each of the two stationary branching-coalescing particle systems on ${\mathbb{Z}}^3$ referred to in Aldous's source. Let $\rho$ denote its particle intensity and let $L^{(\rho)}$ be the intended lineage process. The generator, stationary law, parameterization, lineage sampling rule, and direction of genealogical time are not included in the record.

Write $\overline L^{(\rho)}$ for the source's operation of forgetting spatial positions and suppressing short parent-daughter jitter, and let $a(\rho)$ be the prescribed time-rescaling factor. Neither that operation nor the factor is given. The requested conclusion is finite-dimensional-distribution convergence to the specified Poisson split-and-merge lineage process $L$: \[ \left(\overline L^{(\rho)}_{a(\rho)t_1},\ldots, \overline L^{(\rho)}_{a(\rho)t_k}\right) \ \Longrightarrow\ (L_{t_1},\ldots,L_{t_k}) \qquad(\rho\downarrow0) \] for every finite set of times $0\le t_1<\cdots<t_k$ and for both source models, with the intended state space and initialization.

The limiting lineage state, splitting and merging mechanisms, rates, Poisson independence rules, and initial law are also absent. Thus the display records the exact type of convergence asked for but does not supply a guessed split-merge generator or a particular inverse-density time scale. The source asks only for finite-dimensional distributions, not an unannounced stronger path-space convergence theorem.

Short English statement

Prove the low-intensity lineage limit for each of Aldous's two stationary models on the three-dimensional integer lattice, after the specified time change and removal of spatial and short-jitter information. The models, rescaling, and limit generator must be recovered from the original source.

The supplied summary calls this open and lists related lineage work, but is uncertain about the underlying paper's authorship and date and ends mid-comparison. Those other models do not establish this exact limit. The OPEN-TRIAGE label is retained with current primary-literature audit required.

Sources

• Local UnsolvedMath registry, permanent ID 9700005, source code catalogue entry, "Low-density lineage limit of coalescing branching random walk," proposed by David Aldous. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• David Aldous, Open problems, cbrw.html, Problem A, index https://www.stat.berkeley.edu/~aldous/Research/OP/index.html; HTML-text extraction with linked primary-PDF text, accessed 2026-07-29. The exact primary PDF is not bibliographically identified.
• The summary tentatively names "Branching and coalescing particle systems" and possible Burdzy collaboration/circa-1999 dating; that uncertain attribution is not completed or accepted as a verified citation.
• Related titles supplied without arXiv identifiers: "Ancestral lineages for a branching annihilating random walk"; "Pair coalescence times of ancestral lineages of two-dimensional logistic branching random walks" (2024); "Quenched CLT for ancestral lineages of logistic branching random walks" (2024); "Coalescing directed random walks ... converge to the Brownian web." Their comparison is truncated.
• Provenance records a maintained index of 23 active pages; current primary-literature status audit and redistribution terms NEEDS_REVIEW, separately from dataset attribution.