Distinct Higher-Point Motions of Perturbed Arratia Flows

Definitions and mathematical statement

Let $K^{\theta}=(K^{\theta}_{s,t})_{s\le t}$ be the stochastic flow of probability kernels obtained from the source's $\theta$-joining of the Arratia flow, and let $L=(L_{s,t})_{s\le t}$ be the Le Jan--Raimond sticky flow to which the source compares it. Their construction and the matching of their parameters must be recovered from the report. Write $S$ for their common state space. For a flow $K$, its $n$-point transition kernel is \[ P^{K,n}_{s,t}(x_1,\ldots,x_n; A) ={\mathbb{E}}\!\left[\left(\bigotimes_{j=1}^n K_{s,t}(x_j,\cdot)\right)(A)\right], \qquad A\subseteq S^n\ \text{measurable}. \] This definition uses the same random kernel in all coordinates and averages over the flow randomness. Agreement of $n$-point motions means equality of the resulting transition families for all relevant times and initial configurations.

The conjecture is that, for the two flows with the parameters intended in the report, \[ \forall n\in{\mathbb{N}},\ n\ge4:\qquad P^{K^{\theta},n}\ne P^{L,n}, \] and consequently their laws as stochastic flows of kernels are different. Here inequality means that, for each such $n$, some admissible $s<t$, initial vector $x\in S^n$, and measurable $A\subseteq S^n$ distinguish the transition probabilities. It does not assert inequality for every initial vector or every test set.

According to the supplied literature assessment, the report establishes agreement of the two-point motion, whereas disagreement from four points onward is conjectural. No conclusion about the three-point motion is supplied. The record does not define the joining, the sticky-flow parameters, the state-space conventions, or the admissible range of $\theta$; it also does not state whether the claim is to hold for every admissible $\theta$ or only specified nondegenerate parameters. These qualifications remain unresolved and prevent an unconditional parameter-uniform formulation.

Short English statement

The two constructions agree at the level of two-point motion according to the source. The conjecture asks whether every motion involving at least four sampled particles distinguishes their transition laws, and hence distinguishes the flows themselves. The joining and parameter matching still need explicit definitions.

Sources

• Local UnsolvedMath registry, permanent ID 30000227, code catalogue entry, "Distinct Higher-Point Motions of Perturbed Arratia Flows." Proposed year 2005; no proposer is identified.
• Stochastic Analysis and Non-Classical Random Processes (2006), Oberwolfach Reports, pp. 1231--1298, DOI 10.4171/owr/2005/23. Original Oberwolfach Report 23/2005: https://doi.org/10.4171/owr/2005/23. The citation's displayed year 2006 differs from the DOI year and proposed year 2005.
• Supplied literature reference: Le Jan--Raimond, "flows, coalescence and noise," https://doi.org/10.1214/009117904000000207.
• Supplied literature reference: Howitt--Warren, "stochastic flows of kernels," https://arxiv.org/abs/math/0611292.
• Supplied statement status corrected_verified: its verification note says the official report was checked directly and the two formerly unnamed flows were identified from the surrounding construction. This draft does not independently verify that claim.
• Literature triage checked 2026-08-21 labels the problem open and says the checked flow literature does not resolve this exact conjecture. The record was created 2026-08-25 and updated 2026-09-16; the assessment and cited claims require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; report and literature provenance remain as supplied. Statement-only draft awaiting mathematical review.