Randomized-Transport Characterizations of Palm Measures

Definitions and mathematical statement

The source asks whether externally randomized, invariant transports give a characterization of Palm versions through mass-stationarity, and whether transport kernels alone suffice in that characterization.

An allocation sends each location $x$ to one location $\tau(\omega,x)$. A transport kernel allows mass splitting: $T(\omega,x,\cdot)$ is a probability measure on destination locations, measurably depending on the configuration $\omega$ and $x$. Allocations are the special case $T(\omega,x,\cdot)=\delta_{\tau(\omega,x)}$. Invariance means equivariance under translations of both the configuration and the locations. For source and target random measures $\xi,\eta$, a balancing kernel satisfies \[ \int T(\omega,x,B)\,\xi(\omega,{\,\mathrm{d}} x)=\eta(\omega,B) \] for the relevant measurable destination sets $B$. External randomization means that $\tau$ or $T$ may also use an independent stationary background, rather than being determined by the original configuration alone.

For orientation, in a finite-positive-intensity Euclidean setting, a Palm law $Q$ is the law of the configuration viewed from a typical unit of mass. If $\theta_x$ moves the observation origin to $x$, it is characterized by a Campbell identity of the form \[ {\mathbb{E}}\int_B F(\theta_x\omega)\,\xi({\,\mathrm{d}} x) =\lambda |B|\int F\,{\,\mathrm{d}} Q, \] for stationary configurations of intensity $0<\lambda<\infty$, suitable nonnegative tests $F$, and finite-volume $B$. This is an illustrative standard setting, not a restriction silently imposed on the source's general question.

The requested characterization would identify exactly which candidate rooted laws are Palm versions by the intended mass-stationarity invariance tests after the permitted randomization. The record does not define those tests, their kernel class, the acting group, admissibility and intensity assumptions, or what it means to narrow the characterization to kernels alone. Thus the exact necessary-and-sufficient criterion cannot be reconstructed here. In particular, arbitrary kernels are not automatically mass-preserving, and reported equal-intensity/mutual-singularity transport results are not substituted as universal hypotheses or as a complete answer.

Short English statement

Can independent stationary randomization or invariant mass-splitting kernels make mass-stationarity characterize Palm versions of stationary processes, and can the test class be reduced to transport kernels alone? The precise invariance tests are missing.

Sources

• Local UnsolvedMath registry, permanent ID $30001084$, source code catalogue entry, "Randomized-Transport Characterizations of Palm Measures." No individual proposer is supplied.
• New Perspectives in Stochastic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/47, pp. 2655--2702: https://doi.org/10.4171/owr/2008/47.
• Supplied reference, "Last--Thorisson, invariant transports": https://arxiv.org/abs/0906.2062.
• Supplied reference, "Last--Tang--Thorisson, transport kernels": https://www.columbia.edu/~wt2319/Transport%20measure.pdf.
• Supplied literature triage, checked 2026-08-21: status "partially solved." It reports externally randomized invariant transport-kernel constructions and sufficiency under equal intensity and mutual singularity by Last--Tang--Thorisson, resolving broad cases but not every general allocation formulation in the passage. This is supplied dated curation, not independently verified; time-sensitive status, exact hypotheses, and cited claims require independent verification.