Random valuations

Abstract: A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}<sup>d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $σ$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r&gt;0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.

• The paper provides a comprehensive theory of random valuations, describing their properties and characterizing their Lévy measures under specific conditions, particularly for non-negative infinitely divisible valuations
• Monotone, $\sigma$-continuous, non-negative infinitely divisible valuations with independent increments are equivalently represented as sums of Poisson processes of two-valued atoms indexed by closed convex sets
• These valuations can be further refined to stationary and isotropic forms, and under isotropy, they decompose into sums of germ-grain, flat, and global components, with integrative cylinder–Grassmannian and Crofton-based representations

This paper develops a systematic theory of random valuations—stochastic processes on the family $\mathcal K^d$ of compact convex sets in $\mathbb R^d$ satisfying the valuation identity almost surely—and obtains complete characterisations of non-negative infinitely divisible (ID) valuations under monotonicity, $\sigma$-continuity, and independence of increments. The work by Ilienko and Molchanov (2608.19976) positions random valuations between random measures and general set-indexed processes: unlike measures, the domain is not an algebra (additivity holds only for admissible pairs whose union is convex), so non-negative valuations need not be monotone and no Jordan decomposition exists.

Preliminaries and separability

A random valuation $\Phi$ satisfies $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$ a.s. for admissible $K,L$, with $\Phi(\varnothing)=0$. The authors introduce a separability condition requiring approximation of arbitrary pairs by decreasing sequences from a countable family (rational polytopes) preserving convexity of unions. Two structural facts follow. First, separable random valuations have almost surely additive paths. Second, any random valuation whose realisations are almost surely $\sigma$-continuous ($\varphi(K_n)\to\varphi(K)$ whenever $K_n\downarrow K$) is automatically separable; this relies on an explicit construction showing that outer rational-polytope approximations of admissible pairs can be chosen with convex union.

For pathwise-invariant models the deterministic theory applies directly: if almost all realisations are continuous and translation invariant, McMullen's decomposition lifts to a decomposition into measurable random homogeneous components $\mathbb R^d$0 with $\mathbb R^d$1; combined with Hadwiger's theorem, rigid-motion invariance forces $\mathbb R^d$2 with a random coefficient vector. A complementary construction shows that any locally finite random signed measure $\mathbb R^d$3 on the family $\mathbb R^d$4 of closed convex sets yields a $\mathbb R^d$5-continuous random valuation via $\mathbb R^d$6, where $\mathbb R^d$7.

The Lévy measure of ID valuations

For a non-negative ID valuation, each finite-dimensional vector is ID, and patching the Lévy–Khinchin characteristics over finite subfamilies produces a unique Lévy measure $\mathbb R^d$8 on $\mathbb R^d$9, concentrated (in outer measure) on valuations, together with a deterministic valuation given by the essential infimum. The Poisson representation $\sigma$0 then holds in distribution. A key technical point is that $\sigma$1-continuity is not a finite-dimensional constraint; the authors prove nevertheless that if $\sigma$2 is monotone and a.s. $\sigma$3-continuous, the Lévy measure is supported by $\sigma$4-continuous valuations, using a conditioning argument on compound Poisson components split at level $\sigma$5.

Independence of increments versus independent values

The paper draws a sharp distinction between two conditions. Independent values (independence of $\sigma$6 and $\sigma$7 for disjoint $\sigma$8) collapse the model: a monotone $\sigma$9-continuous ID valuation with independent values has the same finite-dimensional distributions as the restriction of a completely random measure $\Phi$0 on $\Phi$1. This contrasts with set-indexed Lévy processes in the literature, which assume precisely this stronger condition.

Independent increments along nested families are strictly weaker and richer. For ID valuations it suffices to check pairwise independence of increments over nested quadruples, and for monotone valuations only pairs $\Phi$2, $\Phi$3 with $\Phi$4. A notable structural result: a non-negative ID valuation with independent increments and vanishing deterministic part is necessarily monotone—the argument that a sum of independent variables cannot be non-negative unless both terms are exploits the probabilistic setting in a way unavailable deterministically.

The main classification theorem states that a monotone $\Phi$5-continuous ID valuation has independent increments if and only if its Lévy measure is the pushforward of a Borel measure $\Phi$6 on $\Phi$7 under $\Phi$8, subject to

$\Phi$9

Equivalently, $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$0 equals in distribution $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$1 for a Poisson point process $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$2 with intensity $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$3. This settles a random variant of the conjecture from [ilien:mol:vis25] that locally finite $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$4-continuous valuations admit representation $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$5: here $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$6 is a pure-jump completely random measure on $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$7. The authors note that not every deterministic monotone $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$8-continuous valuation arises as the expectation of such an ID valuation, since representing measures may be signed. Uniqueness of $\Phi(K\cup L)+\Phi(K\cap L)=\Phi(K)+\Phi(L)$9 holds given the full finite-dimensional distributions, though not from one-dimensional marginals alone—an explicit counterexample on the line is provided.

Stability and homogeneity

Two distinct scaling notions are analysed: scaling of values ($K,L$0) and dilation of the argument ($K,L$1). In terms of $K,L$2: $K,L$3-stability in argument corresponds to $K,L$4 and forces $K,L$5; $K,L$6-stability in value corresponds to $K,L$7 and forces $K,L$8; $K,L$9-homogeneity couples the two scalings. Necessity proofs recover $\Phi(\varnothing)=0$0 from Lévy measures of finite-dimensional vectors via intersections of the hitting families $\Phi(\varnothing)=0$1.

Stationary and isotropic representations

Stationarity of $\Phi(\varnothing)=0$2 translates into translation invariance of $\Phi(\varnothing)=0$3. Adapting Matheron's classification of translation-invariant measures on $\Phi(\varnothing)=0$4, the Lévy measure decomposes over Grassmannians:

$\Phi(\varnothing)=0$5

where $\Phi(\varnothing)=0$6 is a random compact convex set in $\Phi(\varnothing)=0$7 (the centred base of a cylinder), $\Phi(\varnothing)=0$8 are finite kernels on $\Phi(\varnothing)=0$9, and $\sigma$0 is $\sigma$1-finite on $\sigma$2. Thus stationary models are superpositions of Poisson processes of cylinders with direction spaces of all dimensions $\sigma$3: germ-grain models ($\sigma$4), flat processes ($\sigma$5), and a global component ($\sigma$6). Under isotropy, $\sigma$7 with Haar measure $\sigma$8, and for convex $\sigma$9 the intensity reduces to a linear combination of intrinsic volumes,

$\varphi(K_n)\to\varphi(K)$0

An additional rigidity result: a stationary model that is almost surely simple ($\varphi(K_n)\to\varphi(K)$1 for $\varphi(K_n)\to\varphi(K)$2) must be a stationary pure-jump completely random measure—all cylinder components vanish except translated singletons.

Random McMullen decomposition

Specialising to argument-stable stationary models, stability exponents are restricted to integers: if $\varphi(K_n)\to\varphi(K)$3 is $\varphi(K_n)\to\varphi(K)$4-stable in argument, then $\varphi(K_n)\to\varphi(K)$5, and the Lévy measure is supported by affine Grassmannians $\varphi(K_n)\to\varphi(K)$6,

$\varphi(K_n)\to\varphi(K)$7

so all atoms are $\varphi(K_n)\to\varphi(K)$8-flats. The proof uses the Steiner formula: matching powers of $\varphi(K_n)\to\varphi(K)$9 in $K_n\downarrow K$0 kills all but one intrinsic-volume order.

The culminating result is a distributional analogue of McMullen's theorem. Every stationary isotropic $K_n\downarrow K$1-continuous non-negative ID valuation with independent increments is weakly equivalent (equal one-dimensional distributions) to

$K_n\downarrow K$2

with independent components, where $K_n\downarrow K$3 is built from a Poisson process on $K_n\downarrow K$4 with intensity $K_n\downarrow K$5, $K_n\downarrow K$6 normalised by the Crofton formula $K_n\downarrow K$7, and $K_n\downarrow K$8 is $K_n\downarrow K$9-stable in argument. Equivalently, $\mathbb R^d$00. The equivalence cannot be strengthened to equality of finite-dimensional distributions: the example $\mathbb R^d$01 with a unit ball structuring element and its flat-process counterpart $\mathbb R^d$02 share all one-dimensional distributions (both Poisson with mean $\mathbb R^d$03) yet differ jointly—for distant singletons the first gives independent values while the second gives identical ones. Isotropy is essential here: a stationary model built from a non-centrally-symmetric structuring element is ID, stationary, monotone, with independent increments, but admits no stable-sum representation because argument-stable representations are centrally symmetric in distribution.

Extensions

Two appendices relax the increment condition. Under symmetric-independent increments ($\mathbb R^d$04 independent of $\mathbb R^d$05 for admissible pairs), the Lévy measure is supported by suprema $\mathbb R^d$06 of upper semicontinuous quasi-concave functions—a genuinely larger class than indicator-type atoms. On the moment side, for square-integrable processes whose mean is a deterministic valuation, uncorrelated increments hold if and only if $\mathbb R^d$07 on admissible pairs with $\mathbb R^d$08 a monotone valuation; consequently, square-integrable valuations with independent increments sharing one-dimensional distributions agree in covariance on admissible pairs.

Limitations and open questions

Several restrictions delineate the scope of the results. The full classification requires monotonicity, $\mathbb R^d$09-continuity, and independence of increments simultaneously; general ID valuations without these assumptions are described only through existence of a Lévy measure, and the deterministic Conjecture on representation $\mathbb R^d$10 remains open beyond the planar integer-valued case. The McMullen-type decomposition is stated only up to weak equivalence, and the paper leaves open whether linear combinations of shifted-Poisson valuations $\mathbb R^d$11 are dense in the family of stationary $\mathbb R^d$12-continuous monotone ID valuations with independent increments, in analogy with the deterministic density result for $\mathbb R^d$13. The covariance functional $\mathbb R^d$14 is shown to depend on $\mathbb R^d$15 only for admissible pairs, and its behaviour outside that class is not characterised.

Conclusion

The paper provides a complete Lévy–Khintchine-type description of non-negative ID valuations with independent increments, identifying them as Poisson sums of two-valued atoms $\mathbb R^d$16 indexed by closed convex sets, and derives their stationary, isotropic, and stable refinements through cylinder–Grassmannian and Crofton-based representations. The resulting random McMullen decomposition, valid at the level of one-dimensional distributions with integer-valued stability exponents, connects stochastic geometry (germ-grain and flat processes) with the algebraic theory of valuations, while clarifying that independent values—a common assumption in the set-indexed process literature—reduce the model class to completely random measures.