Wishart Processes with Noninjective Semigroups

Definitions and mathematical statement

Let $H$ denote the infinite-dimensional Hilbert space underlying the source's Wishart equation, let $Q$ denote its diffusion operator, and write $T(t)=e^{tA}$, $t\geq 0$, for the semigroup generated by $A$. Injectivity of $Q$ means $\ker Q=\{0\}$; injectivity of $T(t)$ means $\ker T(t)=\{0\}$. The source does not supply the scalar field, separability assumptions, operator domains, or the precise semigroup hypotheses.

Existence question, subject to the missing equation and hypotheses. For the infinite-dimensional Wishart stochastic differential equation referred to as equation (1) in the source, with injective diffusion operator $Q$, does there exist an unbounded semigroup generator $A$ for which a solution exists for some parameter $\alpha\notin\mathbb N$? Here the admissible parameter range and the convention for $\mathbb N$ must be those of the source equation; neither is specified in the supplied record.

The title and accompanying excerpt identify the unresolved regime as one in which the semigroup fails to satisfy injectivity for all positive times. If this means failure of the property $\forall t>0,\ T(t)\text{ is injective}$, it is expressed by \[ \exists t_0>0\quad \ker T(t_0)\ne\{0\}. \] However, the wording "is not injective for all $t>0$" does not unambiguously distinguish this condition from noninjectivity at every positive time. The supplied clean statement asks for an unbounded generator without explicitly imposing either condition. These formulations therefore require reconciliation with the complete source.

Limits of the formulation. Equation (1) itself is absent from the supplied record. Consequently, its drift and noise terms, initial condition, state space, admissibility conditions on $A$ and $Q$, and required notion of solution cannot be specified here. It is also unclear whether the question concerns a fixed arbitrary injective $Q$ or allows $Q$ to be chosen. The question above preserves the supplied target but is not a complete standalone stochastic existence statement.

Qualified accompanying source claims. The excerpt states that, under $\operatorname{rank}(Q)=\infty$, finite-rank Wishart processes exist if and only if $\alpha\in\mathbb N$, subject to hypotheses omitted from the excerpt. Thus finite-rank existence must not silently replace the unrestricted existence question. A preceding fragment discusses processes in $S(\mathbb R^n)$ of rank at most $k$, says a result is known only for $A\equiv0$ and $Q=\operatorname{id}_{\mathbb R^n}$, and cites [5, Theorem 3.10]; the beginning of that assertion and the identity of reference [5] are missing.

The excerpt also states that, if $T(t)$ is injective for every $t\geq0$, a probabilistically and analytically weak solution of equation (1) is Feller for a minor refinement of the weak-$*$ topology on the space of self-adjoint trace-class operators. It reports using this property to prove existence of a unique limiting distribution for the constructed Wishart process. These are supplied contextual claims with incomplete hypotheses, not assertions established by this draft. Finally, the excerpt separately mentions limited understanding of existence for noninjective $Q$ and $\alpha\notin\mathbb N$, even when $H=\mathbb R^n$; that is a distinct question.

Short English statement

The supplied problem asks whether an infinite-dimensional Wishart equation with injective diffusion $Q$ can admit a solution for a noninteger parameter $\alpha$ when its semigroup generator $A$ is unbounded. The accompanying excerpt points to semigroups that fail an injectivity condition. The equation, admissibility assumptions, initial data, and intended quantifiers are missing, so a fully precise standalone statement requires human review of the complete source. The registry labels the problem open; that label is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30005934$, source problem code catalogue entry, title Wishart Processes with Noninjective Semigroups, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2024 and names no individual proposer.
• Constrained Dynamics, Stochastic Numerical Methods and the Modeling of Complex Systems (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/26, pp. 1445--1516. Supplied authoritative source citation and literature-source label: Oberwolfach Report 26/2024. https://doi.org/10.4171/owr/2024/26. The supplied excerpt references equation (1), which is not reproduced, and [5, Theorem 3.10], whose bibliographic entry is not supplied.
• Infinite-dimensional Wishart processes, supplied literature source, DOI 10.1214/24-EJP1173. https://doi.org/10.1214/24-EJP1173. No authors or further bibliographic details are supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open, states that the 2024 report poses this unbounded-generator, noninteger-parameter question, and states that subsequent infinite-dimensional Wishart-process literature does not establish the requested case. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified and reports that the statement was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; no source visit or independent verification is claimed here.