A mathematical preprint has laid out a structural map for a class of random rules that assign values to compact convex sets. Its central result is that every non-negative infinitely divisible valuation in the stated framework has a unique σ-finite Lévy measure, together with a deterministic valuation. In plain language, the measure describes the random component while the deterministic valuation captures the fixed part. The result is conditional on the mathematical assumptions used in the paper.
This is a theory paper, not a study of people, animals, cells or an observational population. Its objects are random valuations on compact convex sets and associated measures; no empirical sample is reported. A random valuation vanishes at the empty set and, almost surely, obeys a set-addition rule: for an admissible pair of compact convex sets, the value of the union plus the value of the intersection equals the sum of the two individual values.
Building the representation
To build its results, the paper starts with finite-dimensional infinite divisibility and uses the Lévy-Khintchine representation. That step produces compatible Lévy measures and deterministic parts for finite collections of the sets under study, providing the foundation for the broader structural descriptions.
One early consequence is an exact monotonicity test. The valuation is monotone exactly when its deterministic part is monotone and the Lévy outer measure is supported on monotone valuations. The order property is therefore checked in both parts of the representation, rather than inferred from the random measure alone.
Independent changes become geometric jumps
The paper’s most concrete geometric characterization concerns independent increments, meaning changes along nested families are independent under the stated setup. For monotone, σ-continuous, infinitely divisible valuations, that property holds exactly when the Lévy measure is generated by weighted hitting valuations from non-empty closed convex sets. Such a jump contributes a positive weight r when a compact convex set K intersects a closed convex set F, and contributes zero when there is no intersection. The theorem turns an abstract independence condition into a geometric description of allowed jumps.
Under a further zero-background condition, the picture simplifies. With zero deterministic part and zero essential infimum at every compact convex set, a non-negative σ-continuous valuation has the stated infinitely divisible, independent-increment property exactly when it is distributed as a pure-jump completely random measure evaluated on the hitting sets of closed convex sets. A completely random measure has independent values on disjoint sets. In a related result, a σ-continuous monotone infinitely divisible valuation with zero deterministic part and independent values on disjoint sets has the same finite-dimensional distributions as a completely random measure restricted to compact convex sets.
Stationarity narrows the picture
Stationarity brings another layer of structure. For stationary valuations, the translation-invariant Lévy measure is represented through Grassmannians indexed by dimension, random compact convex bases in orthogonal complements, measurable finite kernels and a σ-finite scale measure. The paper calls this a cylinder-Grassmannian representation. It is conditional on the integrability and measurability assumptions in the theorem.
Stability makes the model more rigid. For the non-trivial stationary, σ-continuous models with independent increments and no deterministic part, stability under dilation of the argument requires a positive exponent. The exponent is then restricted to the integers from 1 through the ambient dimension d, and the corresponding Poisson atoms are affine subspaces of dimension d minus that exponent. This is a restriction on the mathematical model, not a pattern measured in data.
A useful decomposition, with a clear limit
In the stationary isotropic setting, the valuation has matching one-dimensional distributions with a sum of independent stable components and a non-negative infinitely divisible degree-zero term. The construction is only weakly equivalent to the original valuation. Equality of finite-dimensional distributions is not generally established, so the result does not guarantee that both constructions agree jointly across every collection of convex sets.
Taken together, the findings are conditional structural representations for mathematical objects, not empirical estimates. Their value lies in describing how particular classes of random valuations can be assembled and constrained under the assumptions used in the paper.
The front matter identifies the manuscript as an arXiv version 2 document in math.PR dated 13 September 2026. The acknowledgment says the work received support from the Swiss National Science Foundation through Grant No. 229505.
Paper data and sources
Original title: Random valuations
Authors: Andrii Ilienko, Ilya Molchanov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text