Infinite-Resolution Random-Surface Question

Definitions and mathematical statement

The supplied record concerns random surfaces generated by an unspecified transformation labelled (2), with truncation level $L$ and smoothness parameter $\alpha$. Write $S_L^{(\alpha)}$ only as a placeholder for these surfaces; the supplied extract does not define them. The figure caption lists \[ \alpha\in\left\{\tfrac12,1,\tfrac32,2,\tfrac52,3,\tfrac72\right\}. \] These are illustrated parameter values, not a stated admissible parameter domain.

The recoverable question is: does the construction yield a well-defined limiting random surface as $L\to\infty$? A precise convergence assertion cannot be formulated from the supplied extract: transformation (2), the underlying random field and its weights, the surface parametrization, and the meaning of a well-defined limit are absent. In particular, neither a mode of probabilistic convergence nor a topology for the surfaces is specified.

The supplied dated assessment says that the report reduces this question to continuity of a weighted spherical random field and provides a continuous modification for $\alpha>1$. Here a continuous modification of a field $X=(X(t))_{t\in T}$ means a field $\widetilde X$ on the same probability space such that \[ \mathbb P\left(\widetilde X(t)=X(t)\right)=1\quad\text{for every }t\in T, \] and whose sample functions are continuous almost surely in the topology of $T$. This definition does not supply the missing index space or field construction. The record also says that the report immediately answers the limit question using continuity and differentiability criteria. Those criteria and their hypotheses are not supplied. Consequently, the claim for $\alpha>1$ is retained as attributed source assessment, and neither a general convergence theorem nor a remaining open parameter range can be extracted reliably.

Short English statement

The question asks whether the truncated random surfaces have a meaningful limit as $L\to\infty$. The construction and convergence criterion are missing. Supplied curation attributes a continuous modification for $\alpha>1$ to the report, but does not identify a precise unresolved statement.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30005933$, title "Infinite-Resolution Random-Surface Question," exact source problem code catalogue entry. The record gives proposed year 2024 and names no proposer.
• Constrained Dynamics, Stochastic Numerical Methods and the Modeling of Complex Systems (2024), Oberwolfach Reports, Report 26/2024, pp. 1445--1516, DOI 10.4171/owr/2024/26. Supplied source and literature link: https://doi.org/10.4171/owr/2024/26. The extract refers to transformation (2) and Figure 2; the transformation itself is not included.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved and attributes to the report a reduction to continuity of a weighted spherical random field and a continuous modification for $\alpha>1$. This is supplied dated curation, not independently verified literature assessment. Its background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field says that the report immediately answers the limit question through continuity and differentiability criteria. The statement-status field nevertheless labels the extracted statement unrecoverable. These supplied qualifications do not establish a precise remaining open problem.