Quasi-Invariant Random Measures and Associated Diffusions

Definitions and mathematical statement

Write $\mathcal M(X)$ for the source's space of Radon measures on $X$, and let $G$ be a group acting on this space. The source does not specify the topology of $X$, the precise convention for Radon measures, or the measurable structure on $\mathcal M(X)$. Consequently, the following definition is conditional on supplying a sigma-algebra $\Sigma$ on $\mathcal M(X)$ for which each action map $T_g:\mu\mapsto g\cdot\mu$, $g\in G$, is measurable. No particular choice of these missing structures is asserted here.

A law of a random measure is a probability measure $P$ on $(\mathcal M(X),\Sigma)$. Its pushforward under $g\in G$ is defined by \[ (T_g)_*P(A)=P(T_g^{-1}(A)),\qquad A\in\Sigma. \] The law $P$ is quasi-invariant under $G$ if \[ \forall g\in G,\qquad (T_g)_*P\sim P, \] where equivalence means mutual absolute continuity; explicitly, \[ \forall g\in G\ \forall A\in\Sigma,\qquad P(A)=0\ \Longleftrightarrow\ P(T_g^{-1}(A))=0. \]

Source question. For a group $G$ acting on $\mathcal M(X)$, which probability laws $P$ are quasi-invariant under $G$ and support an associated analysis, such as a Laplace operator or a diffusion on $\mathcal M(X)$?

The source poses a general research question rather than a fully specified classification problem. It supplies neither a class of spaces and actions to be classified nor a definition of the required associated analysis. In particular, it does not specify the domain or properties of a Laplace operator, the topology and path properties of a diffusion, or how either object must be related to $P$ and the action. The examples of a Laplace operator and a diffusion are illustrative; the source does not require both. These missing qualifications prevent a unique, fully precise mathematical target from being extracted.

Short English statement

The question asks which random-measure laws preserve their null sets under every element of a given group action and also admit useful associated analysis, for example a Laplace operator or a diffusion. Quasi-invariance has a precise conditional formulation above, but the underlying spaces, actions, and analytic requirements remain unspecified in the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002704$, titled Quasi-Invariant Random Measures and Associated Diffusions; exact source problem code catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year $2014$ and identifies no individual proposer.
• Supplied report citation: Dirichlet Form Theory and its Applications (2015), Oberwolfach Reports, pp. 2667--2756, DOI 10.4171/owr/2014/48. Supplied source URL: https://doi.org/10.4171/owr/2014/48. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it supplies no narrower problem location within the report.
• Supplied literature source: Oberwolfach Report 48/2014, https://doi.org/10.4171/owr/2014/48. The exact supplied literature-check date is 2026-08-21. This is supplied dated curation, not independently verified: the assessment labels the broad question open and describes a request for a general theory without a precise class or theorem. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with a supplied claim of manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This is recorded provenance, not independent verification of the report attribution, mathematical formulation, or open status.