Mass-stationarity of diffuse random measures via allocations

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a locally compact second countable Abelian group, written additively, with Haar measure $\lambda$. Let $X$ be a random element of a measurable space with a jointly measurable $G$-action $\theta_s$, and let $\xi$ be a random locally finite Borel measure on $G$, with $0$ in its support almost surely. Define $\theta_s\xi(A)=\xi(A+s)$ and $\theta_s(X,\xi)=(\theta_sX,\theta_s\xi)$. For each relatively compact Borel $C$ with $0<\lambda(C)<\infty$, let $U_C$ be Haar-uniform on $C$, independent of $(X,\xi)$, and conditionally sample $V_C$ from $\xi$ normalized on $C-U_C$: \[\mathbb P(V_C\in A\mid X,\xi,U_C)=\frac{\xi(A\cap(C-U_C))}{\xi(C-U_C)}.\] Mass-stationarity means that these conditional laws are well-defined and \[(\theta_{V_C}(X,\xi),U_C+V_C)\stackrel d=((X,\xi),U_C)\] for all such $C$. Assume $\xi$ is diffuse: $\xi(\{s\})=0$ for every $s\in G$ almost surely. For a measurable map $\pi$ from pairs $(X,\xi)$ to $G$, its equivariant allocation is \[\tau_{(X,\xi)}(s)=s+\pi(\theta_s(X,\xi)).\] It preserves $\xi$ if $\xi(\tau^{-1}(A))=\xi(A)$ for every Borel $A$.

Statement recorded in the supplied source.

Is mass-stationarity of $(X,\xi)$ equivalent to \[\theta_{\pi(X,\xi)}(X,\xi)\stackrel d=(X,\xi)\] for every measurable $\pi$ inducing a $\xi$-preserving allocation? Here allocations depend only on $(X,\xi)$; no independent stationary random field is adjoined. [S2]

Short English statement

For diffuse mass on a general locally compact Abelian group, do all preserving allocation shifts characterize a typical location in the mass?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9900008 (AMR-098-0008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hermann Thorisson, Some Open Probability Problems, author preprint; original Universidad de Buenos Aires copy preserved in Internet Archive capture of 27 March 2024, Section 4, mass-stationarity definition and Theorem 4.2, p. 5; Problem 4.1, p. 6. https://web.archive.org/web/20240327021031id_/http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf.