FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES
Bibliographic record
Abstract
Let [Formula: see text] be the space of non-commutative distributions of k-tuples of self-adjoint elements in a C*-probability space. For every t ≥ 0 we consider the transformation [Formula: see text] defined by [Formula: see text] where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on [Formula: see text]. We prove that 𝔹s◦ 𝔹t= 𝔹s + t, for all s, t ≥ 0. For t = 1, we prove that [Formula: see text] is precisely the set [Formula: see text] of distributions in [Formula: see text] which are infinitely divisible with respect to ⊞, and that the map [Formula: see text] coincides with the multi-variable Boolean Bercovici–Pata bijection put into evidence in our previous paper [1]. Thus for a fixed [Formula: see text], the process {𝔹t(μ)|t ≥ 0} can be viewed as some kind of "evolution towards ⊞-infinite divisibility". On the other hand, we put into evidence a relation between the transformations ⊞tand free Brownian motion. More precisely, we introduce a map [Formula: see text] which transforms the free Brownian motion started at an arbitrary [Formula: see text] into the process {𝔹t(μ)|t ≥ 0} for μ = Φ(ν).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".