---
title: "Reachable noise and existence of operator-valued Wishart processes"
date: 2026-09-22
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22892681
pdf: https://github.com/ipitchford/wishart-reachable-noise/releases/download/v0.1.0-candidate/wishart-reachable-noise-v0.1.0-candidate.pdf
repository: https://github.com/ipitchford/wishart-reachable-noise
archive: https://zenodo.org/records/22892681
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# Reachable noise and existence of operator-valued Wishart processes

## Summary
Covariance models describe how many fluctuating quantities move together. In an infinite-dimensional model, an apparently reasonable stochastic equation may have no solution in the desired space. This candidate identifies the geometry controlling existence for a class of operator-valued Wishart processes: where the noise can travel under the drift.

Even noise entering through one direction can spread through infinitely many directions. The paper proposes a complete criterion for deterministic initial covariance, while keeping a necessary Gaussian path-continuity condition explicit.

## Summary for specialists
Let $S(t)$ be a strongly continuous semigroup on a real separable Hilbert space, $Q$ a bounded positive operator, and $x$ a positive trace-class initial covariance. Define $K=\overline{\operatorname{span}}\{S(t)^*\sqrt QH:t\ge0\}$.

For $0<m=\dim K<\infty$, put $k=\operatorname{rank}(P_{K^\perp}xP_{K^\perp})$. The candidate criterion requires $k<\infty$ and
$$
(\alpha\in\mathbb N_0\ \text{and}\ \operatorname{rank}x\le\alpha)\quad\text{or}\quad\alpha\ge m+k-1.
$$
If $\dim K=\infty$, the zero process is available at $\alpha=0,x=0$. Otherwise existence requires a positive integer $\alpha\ge\operatorname{rank}x$ and a continuous Hilbert-space version of the associated Gaussian Ornstein–Uhlenbeck convolution. For $Q=0$, the solution is deterministic for every real parameter. The stated solution law is unique.

## Technical account
A finite-rank transform argument avoids importing an infinite-dimensional determinant formula under a disputed smoothing hypothesis. A singular-scale noncentral Wishart reduction subtracts the rank of initial covariance outside the scale range from the degree parameter. Reachable-subspace geometry then provides the necessary restrictions.

For infinite reachable dimension, the rank argument uses compression dimension $n+2$ at integer parameter $n$. Dimension $n+1$ would land on an unrestricted boundary and would not justify that inference. A separate Gram-path lifting lemma transfers trace-norm continuity to Gaussian factors. Gaussian factors and an independent residual finite-dimensional Wishart diffusion provide the matching constructions.

The manuscript credits the positive-scale distribution theorem, central singular-scale classification, Kalman/Gramian method and older integer Gaussian partition formulas. Its operator-order proposition concerns the exact printed sufficiency implication in Cox–Cuchiero–Khedher's Theorem 2.1; it does not reject all results of that paper. The cover schematically shows an evolving family of directions, not a numerical simulation.

## Evidence, assurance and limitations
The 13-page manuscript supplies the proposed proof. The exact checker verifies finite block-transform identities, Schur factorization and a controllability Gramian, with three corrupted-formula controls and a dimension-boundary safeguard. Normal and optimized Python agree. These checks do not certify the infinite-dimensional analysis.

This is an unrefereed candidate. Internal model-mediated editorial review and a local rerun of a supplied checker do not establish independent reproduction, external specialist review, journal peer review or formal verification. The dated literature comparison found no full earlier criterion, but historical priority remains unestablished.

The full existence classification concerns deterministic initial covariance. The injective-noise noninteger obstruction extends to random initial data by conditioning. No general random-initial existence theorem, strong-solution theorem, pathwise uniqueness, or pricing-performance benefit is claimed.

## Who should care, and why
| Audience | Potential use | Required qualification |
|---|---|---|
| Stochastic analysts | Inspect a proposed resolution of noninteger and degenerate-noise existence questions | The general-semigroup argument needs unaffiliated specialist review |
| Covariance-model researchers | Check drift reachability and initial outside rank before using a model | Gaussian continuity remains a separate analytic requirement |
| Probability theorists | Examine the arbitrary-real-degree singular-scale reduction | Established positive-scale and Gaussian partition results remain prior work |

## Why the problem matters
Admissibility precedes approximation or calibration: a model must exist in its declared state space before those tasks make mathematical sense. The candidate replaces instantaneous noise rank with the subspace reached over time and shows how initial covariance outside that subspace shifts the parameter threshold. This clarifies model construction without establishing downstream empirical gains.

## How to inspect or reproduce the recorded checks
Extract the versioned archive, install its pinned requirements, and run:
```sh
python3 -m pip install -r requirements.txt
python3 verify_identities.py > /tmp/wishart-replay.json
python3 -O verify_identities.py > /tmp/wishart-optimized.json
cmp identity-results.json /tmp/wishart-replay.json
cmp identity-results.json /tmp/wishart-optimized.json
shasum -a 256 -c MANIFEST.sha256
```
Read the complete proof and cited finite-dimensional theorems separately. The archive also records the supplied review response and the previously corrected dimension-boundary error.

## The most valuable next projects
Obtain unaffiliated specialist scrutiny of the continuity and stochastic construction arguments. Develop convenient Gaussian continuity criteria for useful semigroup classes. Extend full existence to random initial laws using measurable solution kernels, with the additional work stated explicitly.

## What is in the evidence package
The package contains PDF, TeX and Markdown manuscripts; the exact verifier and expected output; claim, source, environment and licence records; internal editorial reports; review responses; and a complete manifest. Original prose and data use CC0; original code uses MIT. Cited papers and supplied third-party review files are not relicensed or redistributed.




## Open directions for follow-up research

- Give useful verifiable conditions for Gaussian path continuity under additional semigroup hypotheses.
- Extend the full existence statement to arbitrary initial laws with measurable solution kernels.
- Obtain unaffiliated specialist scrutiny of the general-semigroup proof and the precise printed-order correction.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:wishart-reachable-noise
- Attempt and metric receipts: ep-attempt:wishart-reachable-noise-publication: published / positive; scope assurance-through-publication; target Internal editorial approval, immutable GitHub and Zenodo candidate assets, and complete Evidence Press page/media/canonical readback.; active forecast 110 minutes (70-180); Fermi components Minor revisions and internal editorial gate: 1 x 25/40/65 minutes low/central/high (Procedural prior; not a measured acceleration claim.); Immutable archive and communication assets: 1 x 20/35/55 minutes low/central/high (Procedural prior; not a measured acceleration claim.); Composite checks and two guarded deployments: 1 x 25/35/60 minutes low/central/high (Procedural prior; not a measured acceleration claim.); positive-signal/closure probabilities 0.95/0.85 within 180 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 36/unknown/unknown/6/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 3; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -74 minutes; ratio 0.3273; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No separately instrumented total; ordinary validation excluded.; deduplicatedModelTokens: Fork-aware runtime counter unavailable.; uncachedInputTokens: Runtime token/cache accounting unavailable.; appended measurement corrections measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Preserve the intake null and retain the explicitly limited terminal lower bound; do not use it as a total rework estimate. (reason: Total rework was not separately instrumented; it is included in root active work. The terminal schema requires a number, so zero denotes only the lower bound of separately timed rework intervals, not zero actual repair effort.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, communication
- Decision object: other — Deterministic-initial existence classification for continuous positive trace-class Wishart processes. Scope: General C0 semigroup, bounded positive Q, weak solutions; Gaussian continuity retained explicitly.
- Reusable methods: Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: assurance, publication
- Semantic bridge: explicit — Finite-rank transforms connect the SDE to singular-scale Wishart laws; reachable subspaces impose parameter restrictions; continuous Gaussian factors and residual matrix diffusions construct solutions. Remaining risks: Infinite-dimensional analytic proof requires specialist judgment.; Finite symbolic replay does not certify the full theorem.; Priority and external validation remain unestablished..
- Human judgement gates: Assess the source-to-claim correspondence and written proof.; Preserve rights, status and priority boundaries.; Publication is authorised; external review is a separate dimension.
- Next assurance action: Inspect and independently reproduce the bounded result; explore extensions separately. External review is not a publication prerequisite.
- Claim ceiling: Unrefereed candidate. No authenticated external peer review, formal verification, historical priority, or empirical pricing benefit is established.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Inspectable covariance-process admissibility classification in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted



## Verification status

Unrefereed candidate. No authenticated external peer review, formal verification, historical priority, or empirical pricing benefit is established.

## References

1. Cox, Cuchiero and Khedher (2024), Infinite-dimensional Wishart processes: equation and source open questions. <https://doi.org/10.1214/24-EJP1173>
2. Graczyk, Małecki and Mayerhofer (2018), A characterization of Wishart processes and Wishart distributions: imported positive-scale theorem. <https://doi.org/10.1016/j.spa.2017.07.010>
3. Mayerhofer (2013), On the existence of non-central Wishart distributions: prior necessary and sufficient conditions. <https://doi.org/10.1016/j.jmva.2012.07.010>
4. Van Perlo-ten Kleij (2004), Contributions to multivariate analysis with applications in marketing: integer Gaussian partition precedent. <https://pure.rug.nl/ws/files/9775162/thesis.pdf>
5. He, Karbach and Khedher (2026), Pricing options on forwards in function-valued affine stochastic volatility models, arXiv v2: application context. <https://arxiv.org/abs/2508.14813v2>