Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms
Why this work is in the frame
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Bibliographic record
Abstract
Abstract Let U be an open set of ℝ n , m be a positive Radon measure on U such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi> supp </m:mi> <m:mo>[</m:mo> <m:mi>m</m:mi> <m:mo>]</m:mo> <m:mo>=</m:mo> <m:mi>U</m:mi> </m:mrow> </m:math> ${{\rm supp}[m]=U}$ , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${(P_t)_{t&gt;0}}$ be a strongly continuous contraction sub-Markovian semigroup on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>U</m:mi> <m:mo>;</m:mo> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${L^2(U;m)}$ . We investigate the structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${(P_t)_{t&gt;0}}$ . (i) Denote respectively by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>D</m:mi> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> ${(A,D(A))}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mover accent="true"> <m:mi>A</m:mi> <m:mo>^</m:mo> </m:mover> <m:mo>,</m:mo> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mover accent="true"> <m:mi>A</m:mi> <m:mo>^</m:mo> </m:mover> <m:mo>)</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(\hat{A},D(\hat{A}))}$ the generator and the co-generator of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${(P_t)_{t&gt;0}}$ . Under the assumption that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mn>0</m:mn> <m:mi>∞</m:mi> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mi>U</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>⊂</m:mo> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mover accent="true"> <m:mi>A</m:mi> <m:mo>^</m:mo> </m:mover>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it