Support of Renormalized Singular Stochastic Equations

Definitions and mathematical statement

Let $u$ denote a solution of a singular stochastic partial differential equation requiring renormalisation, and let $T$ denote the time parameter in the supplied notation $\mathcal{L}_T(u)$. Once a topological solution space $X_T$ and the meaning of this law have been specified, write $\mu_T=\mathcal{L}_T(u)$. For a Borel probability measure $\mu_T$ on $X_T$, define \[ \operatorname{supp}_{X_T}\mu_T =\{x\in X_T: \mu_T(U)>0\text{ for every open neighbourhood }U\text{ of }x\}. \] Problem. Characterize this support in terms of solutions of the associated renormalised controlled equations. In particular, identify the admissible controls and renormalisation parameters, the controlled equations, and the topology in which a support characterization is to hold. If $\mathcal{C}_T\subseteq X_T$ denotes the resulting set of controlled solutions, the intended type of characterization can be expressed schematically as \[ \operatorname{supp}_{X_T}\mathcal{L}_T(u) =\overline{\mathcal{C}_T}^{\,X_T}. \] This equality is a formulation template, not an asserted theorem or an exact formula supplied by the record. The record does not specify the SPDE class, noise, spatial domain, initial or boundary conditions, renormalisation prescription, admissible control subgroup, solution concept, or topology. It also does not specify whether $\mathcal{L}_T(u)$ is a terminal-time law or a trajectory law, or how finite lifetimes are handled. Consequently, neither $X_T$ nor $\mathcal{C}_T$, nor the scope of quantification over equations, can be fixed from the supplied evidence.

Short English statement

Describe which solutions can be approximated, in the intended topology, by solutions of suitable renormalised controlled equations, so as to characterize the support of the stochastic solution law. The equation class, controls, renormalisation parameters, and topology remain unspecified. The supplied record labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004573$, source problem code catalogue entry, title Support of Renormalized Singular Stochastic Equations; collection: Oberwolfach Reports Open Problems. The proposed year is recorded as $2020$; no individual proposer is supplied.
• Supplied report citation: New Directions in Rough Path Theory (2021), Oberwolfach Reports, pp. 1955--2019, DOI 10.4171/owr/2020/40. https://doi.org/10.4171/owr/2020/40. The background describes this as a workshop report containing open-problem discussions; no problem-specific page or speaker attribution is supplied.
• Supplied literature source: Hairer and Schonbauer, "support of singular SPDEs." https://arxiv.org/abs/1909.05526. This is the label supplied by the record; a theorem number and full bibliographic citation are not provided.
• Supplied literature source: "OWR 2020/40 report." https://doi.org/10.4171/owr/2020/40.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and states that Hairer and Schonbauer characterize support through controlled solution sets for a broad singular-SPDE class, while the control subgroup and full support description are not explicit in general. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its accompanying qualification says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This registry label does not establish mathematical correctness or verification by this draft.